injection-locked laser network for solving a …xp446hc0861/kai... · 2013-06-17 ·...

INJECTION-LOCKED LASER NETWORK FOR SOLVING

NP-COMPLETE PROBLEMS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Kai Wen

December 2012

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/xp446hc0861

© 2012 by Kai Wen. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/xp446hc0861

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Yoshihisa Yamamoto, Primary Adviser


Daniel Fisher


Hideo Mabuchi

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

NP(Nondeterministic Polynomial)-complete problems are of great interest in various

scientific and engineering fields. Although no algorithm has been found so far to

find exact solutions to NP-complete problems efficiently, many have been discovered

to search for approximate solutions in polynomial time. Quantum computation can

solve problems such as factorization efficiently. However, it still requires exponential

time to solve NP-complete problems because it is limited by various theoretical and

practical factors, especially the requirement of a closed unitary system. This disser-

tation proposes a novel machine, based on an injection-locked laser network, which is

capable of solving NP-complete Ising problems faster and more accurately.

A proposed machine contains one master laser andM slave lasers, and implements

the Ising Hamiltonian ofM spins by injection from the master laser and mutual injec-

tion among slave lasers. For many Ising problems, the laser network spontaneously

finds the polarization configuration with the minimum gain and loss, which is the

ground state of the Ising Hamiltonian. The population of the excited states is de-

pleted exponentially due to their finite loss difference to the ground state. The so-

called minimum gain principle is confirmed both theoretically and numerically. Since

the laser network operates in a highly dissipative system, it is more robust against

noise and loss and more efficient compared to quantum computers which rely on a

closed unitary system.

Further research reveals that the injection-locked laser network has difficulties in

solving Ising problems of larger scales that contain frustrated spin configurations and

degenerate ground states. Because the readouts of the spin values are continuous

v

rather than discrete variables, a false loss landscape is created in the evolution de-

parting from the correct loss landscape generated by the desired Ising Hamiltonian.

Subsequently, the minimum gain principle erroneously drives the laser network to the

incorrect minimum of the false loss landscape.

A self-learning algorithm is introduced to address the false loss landscape. In

each iteration we execute four techniques to predict the correct configurations of

problematic spins and enhance the signal-to-noise ratios. We instruct the laser net-

work according to the predictions and enhancements, and then evolve the system until

steady state is achieved to verify whether it follows the instructions self-consistently.

Finally, the self-learning injection-locked laser network exhibits significantly bet-

ter performance than existing algorithms based on the benchmark results for two

NP-complete subsets of the Ising problems. The laser network can solve all Ising

problems on cubic graphs of up to 20 spins with a small increase in time and fi-

nite success probability. The laser network can further obtain better solutions in

much shorter times than the state-of-the-art approximation algorithm in solving the

two-layer lattice problems of up to 800 spins.

vi

Acknowledgments

I have been extremely fortunate to work on this dissertation with the invaluable help

and support of many people throughout my time at Stanford University. Without

them, it is impossible for me to complete it.

I would first like to thank my PhD advisor, Prof. Yoshihisa Yamamoto, for giving

me the opportunity to work in his group on such fascinating topic. He is a very nice

and intelligent person, and his guidance, experience, and intuitions about physics

have helped provide many insightful ideas throughout the research process. I would

also like to thank Prof. Hideo Mabuchi and Prof. Daniel Fisher, who are also in my

reading committee. My work has benefited greatly from the fruits of their research

achievements, particularly on the physics of lasers and Ising models.

Secondly, I would like to thank my collaborators Kenta Takata and Shoko Ut-

sunomiya from National Institute of Informatics (NII) in Japan, and Zhe Wang from

Stanford University, on the research of the injection-locked laser network. Their en-

gaging discussions and collaborations have helped me clarify many issues on the topic,

debug the programs, and prepare the papers. I would also like to thank Tim Byrnes

from NII, and Georgios Roumpos and Michael Lohse from Stanford University for

the collaboration on the project of open dissipative adiabatic quantum computation.

I am also very thankful to Kiyoshi Tamaki and Hiroki Takesue from NTT Basic Re-

search Lab in Japan, and Qiang Zhang from Stanford University for the collaboration

on the project of differential phase shift quantum key distribution.

Furthermore, I would like to thank all other members in Yamamoto group, who

are very nice to me and have been always willing to help me with my research and

life at Stanford: Na Young Kim, Kaoru Sanaka, Chandra Natarajan, Shelan Tawfeeq,

vii

Jung-Jung Su, Kristiaan De Greve, Darin Sleiter, Peter McMahon, Wolfgang Nitsche,

Leo Yu, N. Cody Jones, Shruti Puri, Katsuya Nozawa, Thaddeus Ladd, Tomoyuki

Horikir, Katsuya Nozawa, Bingyang Zhang, Parin Dalal, Lin Tian, Michael Fraser,

Susan Clark, David Press, Shinichi Koseki, Neil Na, and Kai-Mei Fu.

My gratitude is also owed to Yurika Peterman, Rieko Sasaki, and all administrative

persons in the department of Electrical Engineering and the Ginzton Laboratory, for

preparing documents, organizing events and meetings, and keeping the group, the

lab, and the building running smoothly.

Many thanks to all of my friends at Stanford. Studying and living here at Stanford

with them over the past five years has been one of my best time in my life. I have

been very fortunate to meet Yuankai Ge and Zhi Li for discussions, projects, and

activities altogether. I am thankful for a great deal of help in my study and research

from Zheng Wen, Zizhuo Wang, Jingyu Cui, Tianshi Gao, Chunyan Wang, Yijie Huo,

Saihua Lin, Hai Wei, and Su Chen. I also thank Xiaoyu Liu and Daniel Chang for

proof reading the dissertation. In addition, I enjoyed my time with Wei Wei, Chen

Peng, Chen Wu, Hongyi Zeng, Shuang Li, Dong Liang, Jie Wu, Xukai Shen, Yuchao

Song, Yichuan Ding, Yu Wu, Jiale Liang, Chenchen Wang, Chenyu Wang and all

other friends from Stanford. I really appreciate my friendship with those people.

Finally, I would like to thank my Mom and Dad who are very proud of me and

care about me even though they are on the other side of the ocean. Most importantly,

I would like to express my special thanks to my wife, Jing Yang, for her unconditional

love and support, which have been the strongest motivation for me to move forward

in my life at Stanford.

viii

Contents

Abstract v

Acknowledgments vii

1 Introduction 1

1.1 NP-complete problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Algorithms to solve NP-complete problems approximately . . . . . . 4

1.2.1 Approximation algorithms . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Local search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Survey propagation . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Grover’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 Adiabatic quantum computation . . . . . . . . . . . . . . . . 13

1.4 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Ising problems 18

2.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 NP-completeness of the Ising problems . . . . . . . . . . . . . . . . . 20

2.3 Maximum cut problems . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Equivalence between Ising problems and maximum cut problems 23

2.3.2 Semidefinite programing as a classical approximation algorithm 25

3 Injection-locked lasers 29

3.1 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

ix

3.1.1 Equations of motion for a laser . . . . . . . . . . . . . . . . . 31

3.1.2 Minimum gain principle for single-mode lasers . . . . . . . . . 34

3.2 Injection-locked lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Injection-locked laser network 40

4.1 System design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.1 Overall design . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.2 Spin representation by a slave laser . . . . . . . . . . . . . . . 42

4.1.3 Initialization by the master injection signal . . . . . . . . . . . 44

4.1.4 Implementing the Zeeman terms . . . . . . . . . . . . . . . . . 46

4.1.5 Implementing the Ising coupling terms . . . . . . . . . . . . . 47

4.2 Theoretical model for the injection-locked laser network . . . . . . . . 49

4.2.1 Model with rate equations . . . . . . . . . . . . . . . . . . . . 49

4.2.2 Model with amplitudes and phases . . . . . . . . . . . . . . . 55

4.3 Minimum gain principle . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Advantages of an open dissipative system . . . . . . . . . . . . . . . . 66

4.5.1 Effective loss rate for a non-solution state . . . . . . . . . . . 67

4.5.2 Closed unitary system vs open dissipative system . . . . . . . 71

5 Self-learning algorithm 74

5.1 Limitation of the injection-locked laser network . . . . . . . . . . . . 74

5.1.1 Examples in which the injection-locked laser network fails . . 74

5.1.2 False loss landscape . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Self-learning algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.1 Connected zero spins . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.2 Single isolated zero spins . . . . . . . . . . . . . . . . . . . . . 88

5.2.3 Parity check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.4 Signal-to-noise ratio improvement . . . . . . . . . . . . . . . . 93

5.2.5 Self-learning algorithm . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

x

6 Benchmarking results 106

6.1 Simulation parameters and methods . . . . . . . . . . . . . . . . . . . 106

6.2 MAX-CUT-3 problems . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3 Two-layer lattice problems . . . . . . . . . . . . . . . . . . . . . . . . 111

7 Conclusions 117

7.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A Derivation of the effective loss landscape 121

Bibliography 124

xi

List of Tables

1.1 Comparison of various techniques in solving NP-complete problems. 16

6.1 Summary of the numerical simulation results on the simple MAX-CUT-

3 problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Summary of the numerical simulation results on the two-layer-lattice

problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.1 Summary of the injection-locked laser network and other techniques in

solving NP-complete problems. . . . . . . . . . . . . . . . . . . . . . 119

xii

List of Figures

1.1 Behaviors of various polynomial and exponential time complexities. . 2

1.2 Classification of computational problems if (a) P = NP or (b) P = NP. 3

1.3 The diagram of a local search. . . . . . . . . . . . . . . . . . . . . . 6

1.4 The example graph constructed for solving a 3SAT problem. . . . . . 9

1.5 The geometric representation of a Grover’s iteration. . . . . . . . . . 13

1.6 Time-dependent energy diagram in the adiabatic quantum computa-

tion for solving a Grover’s search problem. . . . . . . . . . . . . . . . 14

1.7 Minimum bandgaps in adiabatic quantum computation for solving

Grover’s problems with various problem sizes. . . . . . . . . . . . . . 15

2.1 An Ising model on a 1D horizontal lattice with nearest-neighbor inter-

actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 An Ising model on a 2D square lattice with nearest-neighbor interac-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 A 3D Ising problem which cannot be embeded in a 2D plane. . . . . 22

2.4 A sample MAX-CUT problem. . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Relaxation of an MAX-CUT problem to use higher dimensional vectors. 25

2.6 Two vectors obtained from SDP is cut by a random hyperplane. . . . 27

3.1 Systematic diagram of a laser. . . . . . . . . . . . . . . . . . . . . . 30

3.2 Loss landscape and gain bandwidth for a single-mode laser. . . . . . 35

3.3 Systematic diagram of a slave laser being injected by the external field

F0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xiii

4.1 Overall design of a laser network onto which an arbitrary Ising model

with 4 spins can be mapped. . . . . . . . . . . . . . . . . . . . . . . 41

4.2 An arbitrary Ising model with 4 spins. . . . . . . . . . . . . . . . . . 42

4.3 A slave laser that represent a spin in an Ising problem. . . . . . . . . 43

4.4 The setup for injecting the master laser’s signal to a slave laser. . . . 45

4.5 Demonstrate of the initial state of a slave laser in a Poincare sphere. 45

4.6 Mutual injection between two slave lasers. . . . . . . . . . . . . . . . 48

4.7 The Poincare sphere showing the |R⟩, |L⟩, |H⟩, |V ⟩, and |D⟩, |D⟩bases along 3 axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.8 The evolution of each slave laser. . . . . . . . . . . . . . . . . . . . . 61

4.9 The initially flat loss landscape for every mode configuration. . . . . . 62

4.10 The loss landscape generated by the Ising Hamiltonian after the com-

putation starts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.11 The loss and gain of the injection-locked laser network at steady state. 64

4.12 An Ising problem with 8 spins. . . . . . . . . . . . . . . . . . . . . . 65

4.13 Simulation results of the amplitudes for solving the Ising problem given

in Fig. 4.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.14 Simulation results of the phases for solving the Ising problem given in

Fig. 4.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.15 Simulation results of the photon numbers in circular polarizations for

solving the Ising problem given in Fig. 4.12. . . . . . . . . . . . . . . 67

4.16 Simulation results of the total photon number populations in each slave

laser for solving the Ising problem given in Fig. 4.12. . . . . . . . . . 68

4.17 Simulation results of the carrier numbers in each slave laser for solving

the Ising problem given in Fig. 4.12. . . . . . . . . . . . . . . . . . . 68

4.18 Simulation results of the spin values measured from the output of each

slave laser for solving the Ising problem given in Fig. 4.12. . . . . . . 69

4.19 The ground state and the first excited state of the injection-locked laser

network at steady state. . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.20 Simulation of the injection-locked laser network in solving 1D Ising

problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xiv

4.21 Simulation of the Grover’s algorithm in solving 1D Ising problem. . . 73

5.1 The simulation result of an example Ising problem in which the injection-

locked laser network ends up with zero spins. . . . . . . . . . . . . . 75

5.2 The graph representation of the Ising problem simulated in Fig. 5.1. 76

5.3 The simulation result of an example Ising problem in which the injection-

locked laser network ends up with an excited state. . . . . . . . . . . 77

5.4 The graph representation of the Ising problem simulated in Fig. 5.3. 78

5.5 The false loss landscape for the three modes with the lowest loss ob-

tained from the simulation in Fig. 5.3. . . . . . . . . . . . . . . . . . 82

5.6 The three most significant joint photon number populations generated

by the false loss landscape obtained from the simulation in Fig. 5.3. 83

5.7 A typical pattern for a zero spin pair. . . . . . . . . . . . . . . . . . 85

5.8 An example of fixing the zero spin pair given in Fig. 5.7. . . . . . . 87

5.9 A typical pattern for a single isolated zero spin and the example of

fixing it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.10 A typical pattern for a triangle of three spins that fail the parity check. 92

5.11 A typical example without any frustrated spin configuration that gen-

erates maximum amplitude for the overall horizontal component of the

mutual injection signals. . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.12 The Poincare sphere for the overall injection signal to the center slave

laser in Fig. 5.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.13 A typical example with frustrated spin configurations which results in

smaller amplitude for the overall horizontal component of the mutual

injection signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.14 The Poincare sphere for the overall injection signal to the center slave

laser in Fig. 5.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.15 The flow of the stochastic self-learning algorithm. . . . . . . . . . . . 97

5.16 The initial drive of self-learning steps without Zeeman terms in solving

the Ising problem defined in Fig. 5.2. . . . . . . . . . . . . . . . . . 101

xv

5.17 The 1st iteration of self-learning steps in solving the Ising problem

defined in Fig. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.18 The 2nd iteration of self-learning steps in solving the Ising problem

defined in Fig. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.19 The 3rd iteration of self-learning steps in solving the Ising problem

defined in Fig. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.20 The 4th iteration of self-learning steps in solving the Ising problem

defined in Fig. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.21 The initial drive of self-learning steps without Zeeman terms in solving

the Ising problem defined in Fig. 5.4. . . . . . . . . . . . . . . . . . 104

5.22 The 1st iteration of self-learning steps in solving the Ising problem

defined in Fig. 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.1 Petersen graph as a cubic graph with 10 vertices. . . . . . . . . . . . 109

6.2 The worst computational time for solving the simple MAX-CUT-3

problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3 A two-layer lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4 Simulation results that solve a two-layer lattice problem successfully. 113

6.5 Simulation results that fail to solve a two-layer lattice problem. . . . 114

6.6 The worst computational time of SDP and the laser network for solving

large-scale two-layer lattice problems. . . . . . . . . . . . . . . . . . 116

xvi

Chapter 1

Introduction

1.1 NP-complete problems

Computers are one of the most important inventions of the 20th century. The method

of using a computer to solve a computational problem is known as a computer al-

gorithm. People have been designing various computer algorithms to solve many

optimization problems. The results have substantially improved the productivity and

organization of our society today. As an example, the computer can help travelers

optimize routes and minimize time and cost, as in a GPS. It can also be used to

minimize costs and maximize output for product design and manufacturing.

After decades of research, people have discovered that certain computational prob-

lems are easy to solve, while others are not. In analyzing the theoretical difficulty

of a problem, we are usually more interested in the asymptotic time complexity

when the problem size M grows very large. We employ the big-O notation, namely,

T (M) = O(f(M)), where f(M) is also a function of M , meaning that there exists a

sufficiently large M0 and a number ϵ such that[1]

|T (M)| ≤ ϵf(M), if M > M0. (1.1)

The above notation means that T (M) grows no faster than f(M) asymptotically.

On the contrary, another notation, namely T (M) = Ω(f(M)), or big-Ω notation, is

1

2 CHAPTER 1. INTRODUCTION

defined if T (M) grows faster than f(M) asymptotically. It means that there exists a

sufficiently large M0 and a positive number ϵ such that

T (M) ≥ ϵf(M), if M > M0. (1.2)

With the notation, we are able to classify computational problems into two cate-

gories. These easy problems are generally referred to as those that can be solved with

polynomial time complexity, namely T (M) is a polynomial function. The set of such

easy problems are denoted as “class P”. On the other hand, the problems, to which

solutions with polynomial time complexity are not yet found, are difficult problems.

For example, if the best solution to a problem is exponential, i.e., T (M) = O(2M),

the time required to solve the problem will quickly become much longer than that

for solving a P problem. The behaviors of the exponential time complexity and the

polynomial time complexity are illustrated in Fig. 1.1.

0 20 40 60 80 10010

0

1010

1020

1030

1040

Problem size / M

T

T(M)=M

T(M)=M5

T(M)=M10

T(M)=2M

Figure 1.1: Behaviors of various polynomial and exponential time complexities.

Furthermore, not all computationally difficult problems are equal. People have

defined a special subset of problems called NP-problems, which can be solved by a

non-deterministic Turing machine in polynomial time. As the deterministic Turing

machine is a special case of non-deterministic Turing machines and is the current the-

oretical model of computers, P problems are also a subset of NP-problems, namely P

1.1. NP-COMPLETE PROBLEMS 3

⊂ NP. However, the opposite relationship, i.e. NP ⊂ P remains one of the most impor-

tant open questions in computer science. There also exists a superset of NP-problems

known as NP-hard problems, which are even more difficult than NP-problems.

To address the question of whether P = NP, the concept of NP-complete problems

has been proposed[2, 3, 4]. An NP-complete problem is an NP problem whose solu-

tion in polynomial time will immediately lead for the solutions in polynomial time to

all NP-problems. Therefore, if a solution in polynomial time is found to one of the

NP-complete problems, we will have proven that P = NP, and established that all of

these problems can be solved in polynomial time. It is for this reason that solving an

NP-complete problem in polynomial time has raised substantial interest in the com-

puter science community. Fig. 1.2 illustrates the classification of all computational

problems for the two potential outcomes of “P = NP?”.

Figure 1.2: Classification of computational problems if (a) P = NP or (b) P = NP.

The proof of a problem to be an NP-complete problem is achieved by establishing

the mapping in polynomial time to another NP-complete problem. So far people

have discovered and proven that many important problems are unfortunately NP-

complete. For example, satisfiability(SAT) is the first known NP-complete problem[2].


In addition, the Hamiltonian path problem, traveling salesman problem(TSP), subset

sum problem, and graph coloring problem, etc., are all proven to be NP-complete[3].

These problems have wide applications in many scientific and engineering fields, and

thus the exponential time complexity of current best algorithms becomes a critical

bottleneck in these fields. the computational difficulty generated by the exponential

time complexity from current best algorithms becomes a critical bottleneck in those

fields.

1.2 Algorithms to solve NP-complete problems ap-

proximately

Although at present people have not yet discovered any algorithm to solve NP-

complete problems in polynomial time, it does not mean that we can do nothing

to them. On the one hand, the exponential time complexity may only apply to the

worst case of these problems. On the other hand, in practice, a reasonable good so-

lution, even though not the global optimal solution, is still very useful given limited

computational resources. Such solutions are approximate solutions to NP-complete

problems. Many techniques have been proposed to find the approximate solutions

with time complexity substantially shorter than the exponential time complexity.

Here we only include a very limited list of such techniques.

1.2.1 Approximation algorithms

Approximation algorithms are proven to find approximate solutions with performance

bounds to NP-hard optimization problems in polynomial time [1]. The performance

bounds are often expressed as the performance ratios that characterize how good the

solutions are.

Without loss of generality, finding the performance ratio for an approximation

algorithm is usually achieved by finding a upper bound UB(x) of the global maximum

OPT(x) of a maximization problem x, namely UB(x) ≥ OPT(x). Then we can prove

1.2. ALGORITHMS TO SOLVE NP-COMPLETE PROBLEMS APPROXIMATELY5

that the best solution f(x) obtained by the approximation algorithm is bounded by

ρOPT(x) ≤ ρUB(x) ≤ f(x) ≤ OPT(x) ≤ UB(x), (1.3)

where ρ < 1. In this way, we can prove that

ρOPT(x) ≤ f(x) ≤ OPT(x). (1.4)

A similar method can be used to find the performance ratio ρ > 1 for an approxima-

tion algorithm that solves a minimization problem.

For example, a simple 2-approximation algorithm has been discovered to solve

the traveling salesman problem (TSP) with the restriction that the distances be-

tween cities are Euclidean distances. The basic idea is that the cost of the minimum

spanning tree is the lower bound to the TSP and traveling based on the minimum

spanning tree with shortcuts costs at most 2 times the cost of the minimum spanning

tree, since the shortcuts always cost less in the Euclidean space. Thus the perfor-

mance ratio ρ = 2. Later a better 1.5-approximation algorithm has been discovered

with the same requirement of Euclidean distances[5].

Unlike other algorithms in this section, the performance ratios and the polyno-

mial time complexities of approximation algorithms are rigorously proven. However,

the approximation algorithms are not guaranteed to find the optimal solutions to

NP-complete problems. Therefore, in general, the equivalence between NP-complete

problems cannot make the approximation algorithm that solves one NP-complete

problem approximately subsequently solve other NP problems with good performance

ratios.

1.2.2 Local search

One of the most intuitive way to solve computational problems is local search. In a

general local search the algorithm starts with an initial state and iteratively moves to

a neighbor state that meets certain optimization criteria. We present two examples

of local search algorithms: simulated annealing and genetic algorithm.


From the definition, there are two essential components in a local search algorithm.

The first component is the search space that defines states and their neighborhoods.

The second component is the search criteria, which is more critical. In a local search,

we usually define an optimization function and compare the outputs of the function

on the current state and all of its neighbors. Hence, an intuitive criterion is to move

to the neighbor with the optimal output among all these states, as illustrated in Fig.

1.3. The local search with such criterion takes the name of hill climbing.

However, as shown in the same figure, the intuitive local search usually cannot

find the correct solution. For example, if there exists a local minimum, in which all its

neighbors is worst than itself, the algorithm will unfortunately get stuck in the local

minimum and never find the actual global minimum. The problem of local minima

becomes more severe if the problem is large and consists of complicated structures.

Figure 1.3: The diagram of a local search.

To mitigate the problem of local minima, various more sophisticated local search

algorithms have been proposed. Simulated annealing is one of the most elegant prob-

abilistic heuristic optimization techniques with analogues in physics[6, 7]. The al-

gorithm is inspired from the annealing in metallurgy, consisting of heating and con-

trolled cooling of a material to generate large crystals, reduce defects, and minimize

the energy of the material.

1.2. ALGORITHMS TO SOLVE NP-COMPLETE PROBLEMS APPROXIMATELY7

In particular, the output value of the optimization function is called energy in

simulated annealing. The algorithm contains a randomized process in selecting a

neighbor, characterized by the acceptance probabilities P (E,E ′, T ), in which E, E ′

are the energies of the current state and a neighbor respectively, and T is the tem-

perature. P (E,E ′, T ) is designed in a way that the neighbor with lower energy will

have higher acceptance probability. Nevertheless, for the neighbors with E ′ > E,

P (E,E ′, T ) are still nonzero given the temperature T > 0. If the time is long enough,

the algorithm is able to escape from any local minimum with a finite probability.

Temperature plays an important role in the acceptance probabilities. Physically,

if the temperature is higher, the atoms in a crystal are more likely to jump to states

with higher energies. On the other hand, if the temperature is lower, the atoms prefer

staying in states with lower energies. So P (E,E ′, T ) will have higher probabilities

for all possible E ′, even if E ′ > E at higher T . When T is reduced, P (E,E ′, T )

will become dominated by P (E,E ′min, T ), where E

′min is the minimum energy among

all neighbors of the current state. To simulate the cooling process in annealing, the

temperature is set to a high value and gradually decreases to 0 according to certain

time-dependent function, named the cooling schedule.

The cooling schedule therefore controls how good the final solution is. Intuitively,

if the cooling is too fast, the algorithm resembles hill climbing and is very likely

to be trapped in local minima. If the cooling is slow enough, the physical analogy

assumes that the crystal will go to the ground state with the lowest energy. In fact,

simulated annealing is proven to find the global optimal solution with the probability

approaching 1, if the cooling rate is slow enough[8, 9]. However, the time required

to achieve the global optimal solution is usually impractically long, and people often

use a faster cooling schedule to get an approximately good result.

Genetic algorithm is another local search technique that avoids local minima with

the heuristic inspired by the analogy of natural evolution. The algorithm tries to find

the optimal solution by resembling inheritance, mutation, selection, and crossover

from the natural evolution.[10]

Genetic algorithm is also an iterative process. Each iteration is called a genera-

tion. In a generation, the algorithm usually starts with a number of states, named a


population of individuals, and employs crossover and/or mutation operations to re-

produce offsprings of the current generation. Then the algorithm evaluates the fitness

of each offspring using an optimization function and keeps a number of best offsprings

and current individuals as the next generation.

Unlike simulated annealing which is proven to converge to the global optimal solu-

tion, there is not theoretical foundation on the convergence of the genetic algorithm,

namely, whether they can find a global optimal solution or not. But practically, they

can find very good solution in short time. Particularly, the offsprings can be con-

sidered as the neighbors of the current generation. The selection rule serves as the

method to choose better individuals. The random crossover and mutation not only

provide a way to generate offsprings, but also lead the algorithm to escape from po-

tential local minima. As a result, the genetic algorithm has the potential to find very

good solutions providing that the parameters and methods are properly designed.

1.2.3 Survey propagation

Survey propagation is proposed to solve certain optimization problems, especially

satisfiability(SAT) and graph coloring[11, 12]. Survey propagation has its physical

origin from the spin glasses, a basic and generic model in physics[13].

The basic idea of the survey propagation starts with constructing a graph that

represents aK-satisfiability(K-SAT) problem, whereK denotes the number of literals

in every clause of the boolean formula[14]. The graph G has two sets of verticesX and

A, in which X includes all variables and A includes all clauses. If a clause includes

a literal of a variable, an edge is created to connect the clause and the variable and

the edge weights +1 or -1 depending on whether the variable appears as original or

negative in the clause. Therefore, the edges in the graph G only connect vertices

between X and A, and the degree of each vertex in A is exact K. Fig. 1.4 illustrates

the example graph constructed for solving a 3SAT problem.

To find a solution to the K-SAT, survey propagation method employs a message-

passing procedure. Each edge is assigned a value representing the probability of a

warning sent from the clause vertex to a variable vertex connected by the edge, which

1.3. QUANTUM COMPUTATION 9

Figure 1.4: The example graph constructed for solving a 3SAT problem with theformula F = (x1 ∨ x2 ∨ x3) ∧ (¬x1 ∨ x2 ∨ x4) ∧ (x3 ∨ ¬x4 ∨ ¬x5). The square nodesrepresent the clauses and the circular nodes represent the variables. The solid edgesweigh +1 and the dashed edges weigh −1.

takes the name of “survey”. The procedure includes many iterations, and in each

iteration, the surveys of all edges are updated according to the surrounding surveys

excluding the edge of itself. We can image that during the update of a survey, the

survey itself is removed from the graph and effectively creates a cavity, and all nearby

edges pass their surveys as messages to the current survey in order to find the solution.

Although the convergence of survey propagation is not proven rigorously on gen-

eral graphs other than trees in theory, simulation results have revealed its capability

of finding the correct answers to K-SAT and other problems in average cases. People

also discover connection between survey propagation to other heuristic algorithms

such as warning propagation and belief propagation[15, 12].

1.3 Quantum computation

In the recent decades, a new paradigm of computation has emerged to be a promising

field of solving computational problems[16]. The new paradigm seeks for the power

of quantum mechanics to implement efficient algorithms, and thus the apparatus are

named quantum computers. On the contrary, the convention computers are often


referred as classical computers.

The concept of quantum computing was first introduced by Richard Feynman

in 1982[17]. Afterwards, people have spent enormous efforts in finding quantum

algorithms that could demonstrate its potential advantages. Shor’s algorithm in fac-

torizing large integers is one of such examples which could achieve polynomial time

complexity[18], while the current best classical algorithm still runs in exponential

time[19]. However, the current research still shows that the quantum computers are

limited by various factors in solving general NP-complete problems efficiently[20].

1.3.1 Grover’s algorithm

Grover’s algorithm is one of the most important quantum algorithms that is capable

of solving general NP-complete problems[21]. The algorithm searches for one or more

desired states in an unsorted database more efficiently than the classical algorithm.

The speedup is achieved by the superposition of all possible states in a quantum sys-

tem and the unitary operations which can be applied to the states in the superposition

in parallel.

Firstly, we introduce the concept of a quantum bit, abbreviated as a qubit. The

qubit is the quantum counterpart of a classical bit. A qubit can stay in one of the

two different states |0⟩ and |1⟩. It can also stay in a superposition of the two different

states, namely α|0⟩ + β|1⟩, given that |α|2 + |β|2 ≡ 1. Therefore, if a quantum

system consists of M qubits, the general quantum state which the system can take,

is expressed as

|ψ⟩ =2M∑x=1

αx|x1x2 · · · xM⟩, (1.5)

in which x1x2 · · · xM is the binary representation of x and∑

x |αx|2 ≡ 1[16].

The unique feature of superpositions allows us to store a database of N entries

into a quantum storage with only M = log2N qubits and x labels each entry from 1

to N = 2M . If we initialize each qubit in state |+⟩ = 1√2(|0⟩ + |1⟩), the entire initial


state inside the quantum storage is

|ψ0⟩ =1√2M

(|00 · · · 0⟩+ |00 · · · 1⟩+ · · ·+ |11 · · · 1⟩) . (1.6)

Secondly, we introduce the oracle which is a unitary operation acting on above

quantum system and an ancillary qubit initialized as |0⟩. Suppose that there is only

one desired answer state |x0⟩, the output of the oracle is defined as

O(|x0⟩|0⟩) = |x0⟩|1⟩, (1.7)

O(|x⟩|0⟩) = |x⟩|0⟩, ∀x = x0. (1.8)

If the operation is applied to a superposition defined in Eq. (1.5), the linearity of a

unitary operation will give the following output

O(|ψ⟩|0⟩) =2M∑x =x0

αx|x1x2 · · · xM⟩|0⟩+ αx0 |x0⟩|1⟩. (1.9)

The operation is applied to every component in the superposition simultaneously and

only flips the ancillary qubit when the component is |x0⟩.

In general, the oracle is considered to be a black box in Grover’s search algorithm

because the actual implementation is not specified. But Since the verification of an

answer in an NP problem takes polynomial time, we can assume that the oracle only

requires polynomial time and other resources.

Naively, could we use only the oracle to separate the answer state out of 2M − 1

non-answer states? Unfortunately, the answer is no. Note that the probability of the

finding the ancillary qubit to be in state |1⟩ is |αx0 |2, which is exponentially small,

namely, 1/2M , if |ψ⟩ is initialized in the state given by Eq. (1.6). The exponen-

tially small probability amplitude thus imposes the major bottleneck for a quantum

computer in solving computational problems using the oracle.

Grover’s algorithm is then proposed to mitigate the bottleneck. Basically, it in-

troduces a procedure to amplify the probability amplitude of the answer state. Par-

ticularly, in each iteration, a Grover’s iteration employs two operations: U and G


defined as[21]

U |x0⟩ = −|x0⟩, (1.10)

U |x⟩ = |x⟩, ∀x = x0, (1.11)

G = 2|ψ0⟩⟨ψ0| − I, (1.12)

in which I is the identity operator. The operator U is implemented by the oracle if

the ancillary qubit is prepared as |+⟩ = 12(|0⟩+ |1⟩), namely

O|x0⟩|+⟩ = −|x0⟩|+⟩, (1.13)

O|x⟩|+⟩ = |x⟩|+⟩, ∀x = x0. (1.14)

The operator G can also be implemented efficiently[16].

Then a Grover’s iteration is to apply G and U in a sequence. Mathematically,

the Grover’s iteration is applied to an input state |ψ⟩ and outputs UG|ψ⟩. In the

geometric configuration space shown in Fig. 1.5, the Grover’s iteration can to rotate

the input state towards the answer state. The probability amplitude of the answer

state in the superposition is amplified after each iteration. Grover has proven that the

maximum number of iterations are bounding by O(√N) = O(

√2M). Note that here

O() refers to the big-O notation in time complexity analysis rather than the oracle.

The result means that the Grover’s algorithm could achieve a square root speedup

in searching an unstructured database, while classical computers have no way but

iterate every entry and yield O(N) time complexity.

However, in solving NP-complete problems, the square root speedup is far from

enough and the time complexity of the Grover’s algorithm is yet exponential. More-

over, quantum computers are very vulnerable to the environment and noises and lose

the coherence among different components in a superposition. Strong error corrections

is required for the Grover’s algorithm to calculate correctly[16, 22, 23, 24, 25, 26].


Figure 1.5: The geometric representation of a Grover’s iteration.

1.3.2 Adiabatic quantum computation

Another way of performing quantum computation utilizes adiabatic evolution of a

quantum system, named adiabatic quantum computation[27, 28]. The adiabatic evo-

lution is a time-dependent evolution which has a finite energy difference between the

ground state and excited states. The adiabatic theorem states that the quantum

system always stays in the ground state, providing that evolution is slow enough and

the system is initialized in the ground state[29].

Firstly, we construct the adiabatic evolution to solve an NP-complete problem. We

design a problem Hamiltonian HP in which its lowest eigenvalue, namely, the energy

of its ground state, is the answer to problem. We also find a beginning Hamiltonian

HB whose ground state is easy to prepare. Then by combining the both Hamiltonian

together, the overall Hamiltonian used in the evolution is given by

H(t) = (1− s(t))HB + s(t)HP , (1.15)

in which s(t) is a time-dependent function with s(0) = 0 and s(T ) = 1. The evolution

time starts with t = 0 to t = T . Therefore, the algorithm starts with the initial state

as the ground state of HB and slowly evolves the system Hamiltonian from HB to


HP .

Secondly, to keep the system from being excited, the evolution time should satisfy

T ≫max0≤s≤1

∣∣⟨ψ1(s)∣∣dHds

∣∣ψ0(s)⟩∣∣

g2min

, (1.16)

in which ψ0(s) and ψ1(s) are the instantaneous ground state and the first excited

state respectively, and gmin is the minimum energy difference between the ground

state and the first excited state, namely, the minimum bandgap. Therefore, in order

to find the correct answer, the time for the algorithm is inverse proportional to g2min,

which is the major bottleneck in the adiabatic quantum computation.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

s

Ene

rgy

/ H

Figure 1.6: Time-dependent energy diagram in the adiabatic quantum computationfor solving a Grover’s search problem with problem size M = 4. The blue solid line isthe ground state energy while the green solid line lower than 1 is the first excited state.The minimum bandgap is extracted from the minimum energy difference betweenthem at s = 0.5.

Unfortunately, a difficult computational problem usually yields the minimum

bandgap exponentially small[30]. For example, in solving the Grover’s search prob-

lem, namely, finding an entry |x0⟩ in an unordered database of 2M entries, we can

design the problem Hamiltonian as

HP = I − |x0⟩⟨x0|, (1.17)


and the beginning Hamiltonian as

HB = I − |+⟩⟨+|, (1.18)

in which |+⟩ = 1√2M

(|00 · · · 0⟩+ |00 · · · 1⟩+ · · ·+ |11 · · · 1⟩). It is easy to find that the

ground states of HP and HB are the answer state |x0⟩ and |+⟩ respectively. A simple

design of the adiabatic quantum computation utilizes a linear function s(t) = t/T in

Eq. (1.15). Fig. 1.6 demonstrates the instantaneous energy spectrum for solving a

Grover’s search problem. The minimum bandgap is found at s(t) = 0.5. However,

if we calculate the minimum bandgap for different problem sizes, as shown in Fig.

1.7, it decreases exponentially with M , and results in exponentially long time for the

algorithm to succeed. By choosing an optimal function of s(t), the time complexity

of the adiabatic quantum computation in solving the Grover’s search problem is

O(√2M)[31, 32], which is the same square root speedup as the Grover’s algorithm

discussed in section 1.3.1.

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

M

Min

imum

ban

dgap

Figure 1.7: Minimum bandgaps in adiabatic quantum computation for solvingGrover’s problems with various problem sizes.


1.4 Motivations

As a summary, we have introduced some major techniques in solving the NP-complete

problems, both classically and quantum-mechanically. In Table. 1.1, we compare the

capabilities, requirements, and bottlenecks of these techniques. The takeaway from

the table is used as the guideline in designing our new computational systems.

Table 1.1: Comparison of various techniques in solving NP-complete problems.Physical Optimization Closed/open

Technique system mechanism system Bottleneck

Approximation N/A Problem N/A Problemalgorithms dependent dependentSimulated Crystals Thermal Open Local minimaannealing relaxation dissipative

systemGenetic Biological Selection N/A Local minima

algorithms spices ruleSurvey Spin Message- N/A Graph cycles

propagation glasses passingGrover’s Quantum Grover’s Closed unitary Quadratic increasealgorithm computers rotation system of probability

amplitudesAdiabatic Quantum Adiabatic Closed unitary Minimumquantum computers theorem system bandgap

computation

Firstly, we show that Grover’s algorithm and the adiabatic quantum computation

require a closed unitary system with little noise from the surrounding environment.

Such system is very difficult to implement in practice and usually requires tremendous

resources in error correction[26]. In adiition, a closed unitary system may impose lim-

itations to quantum computers in finding the correct answers efficiently. For example,

it is proven that with the oracle operation, any quantum algorithm using unitary op-

erations cannot do better than the square root speedup in the Grover’s algorithm in a

closed system[20]. Therefore, one of the motivations of our contribution is to exploit

the potentials of an open dissipative system.

Secondly, the major bottlenecks of many classical algorithms usually relate to the

1.4. MOTIVATIONS 17

problem of local minima. These algorithms consist of iterations that search for the

best candidates based on current results. Nevertheless, if the candidates are all worse

than existing results, the algorithms will get stuck and have no clue on whether a

lower value exists. This fact motivates us to propose new mechanisms that drive the

system towards the global optimal solutions more efficiently. We will also explore any

potential bottlenecks for the new mechanisms as comparison.

Chapter 2

Ising problems

Ising problems are the intended targets for our newly-proposed computational sys-

tem. We start with the definition of Ising problems, and then discuss their NP-

completeness. MAX-CUT problems are equivalent to Ising problems with a direct

one-to-one mapping. We also present the state-of-the-art approximation algorithm

that solves MAX-CUT problems. Therefore the approximation algorithm is useful in

demonstrating the performance improvement of our new system, as discussed in later

chapters.

2.1 Problem definition

Ising problems are one of the most important computational problems with a physical

origin, namely, the Ising model. They are named after physicist Ernst Ising who

studied and solved the one-dimensional Ising models in his thesis during the 1920s[33].

The Ising model formulates a mathematical model of ferromagnetism in statistical

mechanics. The particles in the model are called spins which take on one of the two

states: σz = +1 (spin up) or σz = −1 (spin down). A spin usually represents the

magnetic dipole moment of an atom or an electron. Considering a system ofM spins,

18

2.1. PROBLEM DEFINITION 19

the Ising Hamiltonian that describes the system dynamics and energy is written as

H =M∑i=j

Jijσzi σ

zj +

M∑i

λiσzi . (2.1)

The right hand side of Eq. (2.1) consists of two terms. The first term is often

referred to as the Ising term, which describes the two-spin interaction. For exam-

ple, spin i and j could create their own surrounding magnetic field. As a result,

spin j may be affected by the field from i and vice versa if the two spins are close

enough. The Ising coefficient Jij represents the strength of the coupling and is usually

symmetric, namely, Jij = Jji. The second term is referred to as the Zeeman term,

which only includes the energy from single spins. Practically, the Zeeman term can

be implemented by applying external magnetic field on the spins.

Since each spin can take two values, the number of all possible states are 2M , in

which M is the problem size. In statistical mechanics, the probability of the system

being in certain state σ = (σz1, · · ·σz

M) at certain temperature T is given by the

Boltzmann distribution[34]

Pβ(σ) =e−βH(σ)s

Zβ

, (2.2)

where β = 1kBT

as the inverse temperature and kB is the Boltzmannn constant. The

normalization constant is called the partition function, derived as

Zβ =∑σ

e−βH(σ), (2.3)

summing over all 2M possible states.

To analyze the Ising model, we first examine the extreme case when the temper-

ature T approaches 0. In this case, β goes to infinity, and the only possible state

in which the system can reside under Boltzmann distribution is the ground state

whose energy derived by the Hamiltonian has the lowest value. Based on the zero

20 CHAPTER 2. ISING PROBLEMS

temperature solution, people could study the dynamics of any excitation in the sys-

tem. Therefore, searching for the ground state is one of the most important tasks in

studying the Ising model.

An “Ising problem”, then, is to search for the ground state whose energy given by

Eq. (2.1) is the global minimum. The problem is non-trivial. In general, since

the number of states are 2M , the search space is exponentially large for a large

problem size. The Ising model and Ising problems are widely used and studied in

physics[35]. Particularly it relates to spin glasses which have been actively studied

for decades[36]. Moreover, as the Ising problems are optimization problems searching

for global minimum energy, they provide important value resources in many scientific

and engineering fields outside of physics. For example, researchers have shown that

the Ising problems have applications in biological and medical research such as protein

folding[37]. Other applications of the Ising problems include circuit/microprocessor

design[38], channel assignments for wireless communications[39], image segmentation

in computer vision[40, 41, 42], and so on.

2.2 NP-completeness of the Ising problems

As we mentioned in the last section, the Ising problem, as a ground state search

problem for 2M different states, is difficult in general. People had spent decades in

finding solutions to the Ising problems, until Barahona proved that the Ising problems

are NP-complete in 1982[43].

Fortunately, under certain circumstances, the Ising problems are solvable. The

major contribution by Ernst Ising in his thesis was to solve the one-dimensional(1D)

Ising problems analytically[33]. An 1D Ising model is illustrated in Fig. 2.1, in

which all spins line up in 1D, and the Ising interaction coefficients Jij are nonzero

only for the nearest neighbors. It models a linear horizontal lattice with only nearest

neighbor interactions. In particular, if Jij between two nearest spins are negative,

namely, Jij < 0, the two spins tend to share the same sign in the ground state as

such configuration gives a lower energy. Hence, the interaction is called ferromagnetic

interaction. If Jij > 0, named anti-ferromagnetic interaction, the two spins tend to

2.2. NP-COMPLETENESS OF THE ISING PROBLEMS 21

stay with the opposite signs in the ground state in order to generate a lower energy.

Based on the solution to 1D Ising model, Ernst Ising discovered that the cor-

relation of two spins ⟨σzi σ

zj ⟩ decays exponentially in the distance between the two

spins, namely |i− j|, given a finite temperature. The result shows that the system is

disordered, and therefore excludes any phase transition in 1D Ising model.

Figure 2.1: An Ising model on a 1D horizontal lattice with nearest-neighbor interac-tions.

Many years after Ising’s solution on 1D Ising model, people eventually discovered

the solutions to the two-dimensional(2D) Ising model[44, 45]. An 2D Ising model is

illustrated in Fig. 2.2, in which the spins form a square lattice on a 2D plane and

the interactions also only apply to the nearest neighbors. The solution is generally

derived from a transfer matrix. The results show that unlike the 1D Ising model, the

2D Ising model has phase transition.

Figure 2.2: An Ising model on a 2D square lattice with nearest-neighbor interactions.

Since we have mentioned the 1D and 2D Ising models, we present the general

definition of dimensionality in characterizing the Ising models. Here we only consider

the Ising coupling terms, namely, the Ising Hamiltonian is written as

H =M∑i =j

Jijσzi σ

zj . (2.4)

Because Jij connects spin i and j, we can utilize a graph with edges to represent the

model. Particularly, we generate a graph G with M vertices representing M spins.

For any non-zero Jij, an edge connecting vertex i and j is put in the graph and its


weight is Jij. Therefore, we can define the 1D and 2D Ising models as the planar

graphs, in which the graph representations of the models can be embedded in a 2D

plane, with all edges intersecting each other only at their ends.

By defining the dimensionality and planar/non-planar of an Ising model, people

have proven that 3D Ising problems are unfortunately NP-complete[43]. Nonpla-

narity in the graph representation of an Ising problem plays a critical role in its

NP-completeness[46]. Fig. 2.3 presents an example of the Ising problem which can-

not be embedded in a 2D plane. A large-scale Ising problem is usually nonplanar,

which imposes huge difficulties for people to find the global optimal solutions.

Figure 2.3: A 3D Ising problem which cannot be embeded in a 2D plane.

2.3 Maximum cut problems

The original proof of NP-completeness of the Ising problems is complicated. However,

the NP-completeness can be derived more easily from maximum cut (MAX-CUT)

problems. MAX-CUT problems are one of the earliest 21 NP-complete problems[3].

The definition of a MAX-CUT problem is described as follows: Given an undi-

rected graph G with a vertex set V and an edge set E, we would like to find a

separation of vertices that can better cut the edges. Particularly, we divide the ver-

tex set V to two sets S and V \ S, and the cut of the two sets is defined as

CUT(S) =∑

eij∈E,vi∈S,vj∈V \S

wij, (2.5)

2.3. MAXIMUM CUT PROBLEMS 23

in which eij is an edge connecting vertices vi and vj, wij is the weight of the edge eij,

and the summation includes all edges that only connect between vertices in S and

V \ S. Therefore, the MAX-CUT problem is to find a set S such that CUT (S) is

maximized, namely

MAX-CUT(G) = maxS⊂V

CUT (S). (2.6)

Fig. 2.4 exhibits a graph with 4 vertices and 6 unit-weight edges. We can easily

find that the solution is to put any two vertices into the set S and the MAX-CUT of

the graph is 4. If there is only one vertex in the set S, the cut is only 2.

Figure 2.4: A sample MAX-CUT problem with 4 vertices where all edges weigh +1.The cut is labelled by the red vertices and MAX-CUT(G) = 4, including the 4 rededges.

2.3.1 Equivalence between Ising problems and maximum cut

problems

We introduce MAX-CUT problems because it is equivalent to the Ising problem

without Zeeman terms through direct one-to-one mapping[38, 47]. This allows us to

compare the simulation results between our proposed machine and the state-of-the-art

approximation algorithm for solving MAX-CUT problems.

Since the Ising coupling terms can also be represented in a graph G = (V,E), in

which V is the vertex set and E is the edge set, we can define the MAX-CUT problem

on the same graph G and the weight of each edge is equal to the corresponding


Ising coupling coefficient, namely, wij = Jij. Then the MAX-CUT problem tries to

maximize the following quantity,

MAX-CUT(G) = maxS⊂V

CUT(S) = maxS⊂V

∑(i,j)∈E,i∈S,j∈V \S

Jij, (2.7)

where (i, j) is an edge connecting between S and V \ S.

To map the above problem to the Ising problem, we formulate the following Ising

Hamiltonian for the same graph,

H =∑

(i,j)∈E

Jijσzi σ

zj . (2.8)

Since σzi can be either +1 or −1, we can define the following cut

S = vi ∈ V, s.t. σzi = +1, (2.9)

and

V \ S = vi ∈ V, s.t. σzi = −1. (2.10)

Then the Hamiltonian is re-written as

H = −∑

(i,j)∈E,i∈S,j∈V \S

Jij +∑

(i,j)∈E

Jij (2.11)

= −CUT(S) +

∣∣∣∣∣∑ij

Jij

∣∣∣∣∣ . (2.12)

As∣∣∣∑ij Jij

∣∣∣ is fixed, minimizing the Ising Hamiltonian will subsequently find the

MAX-CUT on the same graph, namely,

minH = −MAX-CUT(G) +

∣∣∣∣∣∑ij

Jij

∣∣∣∣∣ . (2.13)


In this way, we establish the equivalence between Ising problems and MAX-CUT

problems. Solving the Ising Hamiltonian will directly lead to the solution to MAX-

CUT and vice versa.

2.3.2 Semidefinite programing as a classical approximation

algorithm

Here we introduce the state-of-the-art approximation algorithm for solving the MAX-

CUT problems and subsequently solving the Ising problems, named semidefinite pro-

gramming (SDP)[48]. This algorithm is of particular interests since we are going to

compare the performance of our proposed computational machine with it in solving

large-scale Ising problems.

The approximation algorithm is originated from the relaxation of a MAX-CUT

problem. We first transform the MAX-CUT problem to the following formulation:

maximize∑

(i,j)∈E

wij

(1− vivj

2

)subject to v2i = 1.

If vi can only take either +1 or −1, it is exactly the MAX-CUT problem. In this

case, the cut S is defined to the set of vertices with vi = +1.

Figure 2.5: Relaxation of an MAX-CUT problem to use higher dimensional vectors.

In a relaxation, we allow the variables vi to be in a higher dimension space, namely,

vi ∈ Rn is a vector. Subsequently, we modify our constraints and goal function to


inner product of two vectors, i.e.,

maximize∑

(i,j)∈E

wij

(1− vi · vj

2

)subject to ||vi|| = 1

vi ∈ Rn.

As an example, Fig. 2.5 illustrates two vectors vi and vj in a 3D sphere with unit

radius which satisfy the constraints.

The above optimization problems with relaxation, though in higher dimension,

turn out to be solvable in polynomial time. The formalism of the problems be-

long to a set of problems with positive semidefinite matrices, called semidefinite pro-

gramming(SDP). SDPs can be solved in polynomial time[49] and the state-of-the-art

method is interior point methods with running time[50, 51]

O(n3.5 log ϵ−1), (2.14)

in which ϵ is an arbitrary small error parameter. Therefore, the optimal solution to

the SDP with relaxation is called SDP(G). It is obvious to obtain that

SDP(G) ≥ MAX-CUT(G), (2.15)

because the MAX-CUT problem is just a special case of the SDP.

Since we are able to solve SDP in polynomial time, the final step to solve MAX-

CUT problems approximately is to reduce the result from SDP in higher dimension

back to the original problem. Note that this reduction is not exact and therefore

only gives approximate solutions to the original problem. The relaxation tells us

that if vi · vj ≈ −1 we should separate the two nodes i and j into two sets. The

idea is discovered by Goemans and Williamson as in their famous random rounding

technique[48]. Particularly, the technique choose a uniformly random hyperplane in

the high dimensional space through the origin and use it to cut the vertices into two

sets. Fig. 2.6 demonstrates a random hyperplane that cuts the two vertices on the


hypersphere into two sets.

Figure 2.6: Two vectors obtained from SDP is cut by a random hyperplane duringrandom rounding procedure. As the two vectors are on different sides of the plane,they belongs two different cuts.

Since the random rounding is a randomized algorithm, we are interested in its

expected value, called GW(G), derived as[48]

GW(G) =∑ij

wijarccos(vi · vj)

π. (2.16)

We can find the minimum value for GW(G) as

GW(G) ≥ αGWSDP(G) ≈ 0.878× SDP(G). (2.17)

Note that the result of random rounding is just a special case of the original MAX-

CUT problem. As a result, we obtain the following relation

0.878× SDP(G) ≤ GW(G) ≤ MAX-CUT(G) ≤ SDP(G), (2.18)

and thus

GW(G) ≥ 0.878×MAX-CUT(G). (2.19)

By the definition of performance ratios in approximation algorithms discussed in sec-

tion 1.2.1, we state that SDP with random rounding is a 0.878-approximation to


MAX-CUT problems. Although in general an approximation algorithm for solving

one NP-complete problem may not apply to another NP-complete problem, the di-

rect mapping between MAX-CUT problems and Ising problems allows SDP to find

approximation solutions to both problems.

The SDP with random rounding is the state-of-the-art approximation algorithm,

in the sense that probably no other polynomial approximation algorithm can do better

than it. This is a widely accepted conjecture, called “unique game conjecture”[52, 53].

The conjecture leads to the claim that getting the degree of (0.878+ϵ)-approximation

for MAX-CUT problems is NP-hard, in which ϵ is an arbitrary small number. There-

fore, it is worth comparing our proposed machine with SDP, and our proposed machine

can demonstrate substantial advantages if it can outperform SDP by getting better

results.

Chapter 3

Injection-locked lasers

The novel machine we propose to solve Ising problems is based on the injection-

locked lasers. The injection-locking mechanism between lasers plays a key role in

implementing the Ising coupling terms. The lasers are operating at highly open

dissipative environment in which the gain and the loss compete with each other.

The competition between the gain and the loss may provide us a new optimization

mechanism to solve Ising problems. The photon field in the laser system is coherent

and the coherence may enable the system to probe different modes simultaneously.

Here we briefly present the theoretical basis of the injection-locked lasers.

3.1 Lasers

Lasers are one of the most important breakthroughs in the 20th century. Fig. 3.1

draws the diagram of a typical single-mode laser. The single-mode laser requires

three essential components: a device that amplifies the internal field, a device that

selects the frequency for the internal field, and a device that provides nonlinear gain

saturation. A typical laser consists of a gain medium that implements both lin-

ear amplification via stimulated and spontaneous emissions and nonlinear saturation

mechanisms via atomic absorption, and a cavity with two mirrors that only allows

certain oscillation frequencies and modes.

The internal field A(t) is built up by the competition between the gain and the

29

30 CHAPTER 3. INJECTION-LOCKED LASERS

Figure 3.1: Systematic diagram of a laser.

loss. The field is amplified by the gain medium when it travels back and forth between

the two mirrors. The generated photons either decay through the cavity loss or is

absorbed by the gain medium. The gain and the loss compete against each other.

Initially the gain is greater than the loss and the internal field builds up exponentially.

After the field amplitude increases substantially, the nonlinear absorption becomes

significant and eventually saturates the gain at steady state.

The cavity is also coupled to the external environment. The internal field decays

via cavity loss, and the lost photons into a lasing mode generate the laser output. On

the other hand, the external vacuum field fL is also coupled to the cavity, as shown

in Fig. 3.1. Affected by the incident vacuum field fL, the output field of the laser

experiences a phase shift and the output field operator r is described as

r =ω

Qex

A− fL, (3.1)

in which Qex is the cavity Q-factor to external loss and ω is the frequency of the

output mode.

3.1. LASERS 31

3.1.1 Equations of motion for a laser

The quantum-mechanical Langevin equation describing the dynamics of the internal

field A(t) is given by

d

dtA(t) = −iωA(t)− 1

2

[ω

Q− ECV (t)

]A(t) +

√ECV (t)fG +

√ω

QfL. (3.2)

The above equation involves the dynamics of the photon field operators denoted by

hats, and the electric population operators designated by tildes. Since the electric

dipole moment is assumed to decay at a rate much faster than the photon decay rate

and the electron population decay rate, we adiabatically eliminate the electric dipole

operator[54, 55].

The term ωQdescribes the photon loss rate in the cavity through an output coupling

mirror, in which ω is the frequency of the injection signal and Q is the quality factor

of the cavity. The internal loss rate is neglected because it is much smaller than the

external loss rate. ECV is the photon emission rate operator for the gain medium

into a lasing mode . fG is the Langevin noise operator for the electric dipole moment,

which is originated from random photon emission and absorption by the gain medium.

fL is the Langevin noise operator for the cavity field, originated from the injection

signal noise including a vacuum fluctuation.

We can obtain the c-number rate equation for photon number from Eq. (3.2).

The photon number operator is defined as

n(t) = A†(t)A(t). (3.3)

Using Eq. (3.2), we are able to derive the quantum-mechanical equation for the

photon number operator as[56, 55]

d

dtn(t) = −ω

Qn(t) + ECV n(t) + ECV + Fn(t). (3.4)

To find the noise power of the noise term Fn(t) for the photon number operator, we


derive its two-time-correlation function as[55],

⟨Fn(t)Fn(t′)⟩ = δ(t− t′)

[ω

Q⟨n⟩+ ⟨ECV ⟩(⟨n⟩+ 1)

], (3.5)

where we assume that the injection signal noise is equal to the vacuum fluctuation

level.

By taking the ensemble average to both sides of Eq. (3.2), we get the c-number

rate equation for photon number as

d

dtn = −

(ω

Q− ECV

)n+ ECV . (3.6)

Note that the noise terms are all averaged out by the ensemble average.

On the other hand, at well above threshold at which the lasers phase is well-

defined, we can derive the c-number amplitude and phase equations for the internal

field. Firstly, by switching from Heisenberg picture to Schrodinger picture, we can

convert the quantum-mechanical Langevin Eq. (3.2) to the laser master equation.

Secondly, the quantum-mechanical Fokker-Planck equation is obtained by using the

Glauber-Sudarshan P (α) representation for the field density operator[57]

ρ(t) =

∫P (α)|α⟩⟨α|d2α, (3.7)

in the laser master equation. Thirdly, with the Kramers-Moyal expansion[58], the

equation of motion for the equation (complex) eigenvalue α can be obtained as

d

dtα(t) =

[G− 1

2

(ω

Q

)− S|α|2

]α(t) +

√GΓα, (3.8)

where G is the linear gain coefficient, ω/Q the cavity photon decay rate, S the satu-

ration parameter. The stochastic noise term satisfies

⟨Γα(t)⟩ = 0, (3.9)

⟨Γα(t)Γα(t′)⟩ = 2δ(t− t′). (3.10)

3.1. LASERS 33

Note that the noise behavior of α(t) does not include the vacuum fluctuation, which

is absorbed in the noise of the basis set of coherent states as |α⟩⟨α|. It is also man-

ifested that the noise term in Eq. (3.8) includes only the dipole moment fluctuation

associated with the gain G. Therefore, an actual measurement result features an

extra fluctuation on top of the noise of α(t) due to the intrinsic quantum noise of

coherent states. Moreover, we express the saturated gain dynamics G−S|α| by ECV

as a function of the c-number carrier number, as shown later.

Finally, we express the (complex) eigenvalue α in terms of the amplitude and

phase, α = Aeiϕ. The resulting equations describing the dynamics of the amplitude

and the phase are given by

d

dtA = −1

2

(ω

Q− ECV

)A+ FA(t), (3.11)

d

dtϕ = Fϕ(t), (3.12)

in which FA(t) and Fϕ(t) are the noise terms for the amplitude and phase respectively.

The equations of motion obtained demonstrate that the phase of a laser is only

driven by the random phase noise. In fact, at well above threshold, the dominant

noise terms for the lasers are the quadrature phase noise due to spontaneous emission

noise[58]. In a slave laser with perfect population inversion, the spontaneous emission

rate ECV i almost equals the cavity decay rate ω/Q, and an emitted spontaneous

photon kicks either the amplitudes A(t) or the phases ϕ(t) of the complex fields

A(t) exp[iϕ(t)] of each diagonal mode. Thus, the average phase noise injection rate

amounts to ω/(2Q). Each photon (with unit amplitude) couples to the coherent field

and changes its phase by ±1/A(t), where the sign is randomly chosen with equal

probability.

The quantum-mechanical rate equation for the total electron number operator

N(t) in a slave laser is obtained as[56, 55],

d

dtN(t) = P − Nt

τsp− ECV n(t)− ECV + FN(t), (3.13)


in which P is an average pump rate, N(t)/τsp includes total spontaneous emission,

τsp is the spontaneous emission lifetime of the gain medium. The noise term Fc(t) for

the electron number operator is also described by the following two-time correlation

function,

⟨FN(t)FN(t′)⟩ = δ(t− t′)

[P +

⟨N⟩τsp

+ ⟨ECV ⟩(⟨n⟩+ 1)

]. (3.14)

Also note that the two noise operators Fn and FN are negatively correlated, namely,

⟨Fn(t)FN(t′)⟩ = −δ(t− t′)

[⟨ECV ⟩(⟨n⟩+ 1)

]. (3.15)

Similar to obtaining the c-number rate equation for photon number, we can obtain

the c-number rate equation for carrier number by taking the ensemble average. The

resulting equation is

d

dtN(t) = P − N(t)

τsp− ECV [n(t) + 1] . (3.16)

3.1.2 Minimum gain principle for single-mode lasers

In a single-mode laser, the gain medium is able to generate photons with a broad

range of frequencies, which usually contains enormous discrete modes (104 ∼ 106)

allowed by the cavity as a frequency selection device. However, only one mode is

amplified and output from the single-mode laser.

The underlying mechanism for the laser to output only a single mode is illustrated

in Fig. 3.2. The bandwidth for the gain medium, drawn in the black line, covers the

span of many cavity modes (the vertical lines in orange and red). Nevertheless, the

cavity also has an intrinsic loss landscape, namely, different loss rates for different

modes, depicted in the blue line. The only mode that can oscillate in the single

mode laser is the mode with the minimum loss in the loss landscape, as shown in

the red vertical line, because the gain is saturated to be equal to the minimum loss.

Otherwise, if a mode with higher loss could oscillate at steady state, the gain would

3.1. LASERS 35

be saturated to the higher loss. Then the mode with the minimum loss would have

the gain greater than its loss and its amplitude would keep increasing and violate the

steady state condition. We thus call this mechanism as the minimum gain principle

for single-mode lasers.

Figure 3.2: Loss landscape and gain bandwidth for a single-mode laser.

In a semiconductor laser, from the result of the minimum gain principle, we can

define the fractional coupling efficiency of spontaneous emission into a lasing mode

as[59]

β =1

M. (3.17)

It describes the fact that only one mode is lasing out of M cavity modes within the

bandwidth of the gain medium.

Therefore, the relation between the gain and the carrier number is written as

ECV = βN

τsp. (3.18)

The definition of the gain then correlates the equations between the internal field and

the carrier number in the previous section.

The fractional coupling efficiency of spontaneous emission is also used to find the


pumping threshold for the lasers. The classical pumping threshold is the pumping

rate that generates the gain approximately equal to the cavity loss, namely

ECV ≈ ω

Q. (3.19)

The pumping threshold is then derived as

Pth,c =1

β

ω

Q. (3.20)

The quantum degeneracy threshold is the pumping rate that drives the internal av-

erage photon number to be 1, namely,

n = 1. (3.21)

The resulting pumping threshold is given by

Pth,q =1 + β

2β

ω

Q. (3.22)

Note that usually β is as small as 10−4 ∼ 10−6 in a semiconductor laser, so the

quantum degeneracy threshold is approximately one half of the classical pumping

threshold, namely,

Pth,q ≈1

2β

ω

Q=

1

2Pth,c. (3.23)

3.2 Injection-locked lasers

Injection-locked lasers are the crucial components for our new proposed machine. The

terms that make the Ising problem computationally difficult are the Ising coupling

terms. In order to implement the Ising coupling terms on a laser system, we employ

the injection-locking mechanism to simulate the interactions between the spins.

A injection-locked laser system usually consists of a master laser and a slave laser.

The output of the master laser is injected into the cavity of the slave laser. The

3.2. INJECTION-LOCKED LASERS 37

Figure 3.3: Systematic diagram of a slave laser being injected by the external fieldF0.

diagram of the slave laser is depicted in Fig. 3.3. Compared to the laser diagram

in Fig. 3.1, an external field F0 from the master laser is injected into the cavity of

the slave laser. The master laser is usually operating at well above threshold and its

output field is a coherent state[57]. We can instead use a c-number F0 to represent

the injection field operator F0, namely,

F0 = ζ

√ω

Q

√nMe

iϕ0 , (3.24)

in which nM is the average photon number of the master laser’s internal field, ω/Q

is the cavity photon decay rate of the master laser and√ω/Q is the output coupling

efficiency, ζ is the attenuation coefficient of the injection signal from the output of the

master laser, and ϕ0 is the phase difference between F0 and the slave laser’s internal

field. Note that we use the same cavity photon decay rate for both the master laser

and the slave laser for simplicity.

The injection locking mechanism states that if the frequency of the injection field

falls within the locking bandwith, the frequency of the internal field of the slave laser

is locked to the frequency of the injection field. The locking bandwidth is defined

as[60, 61]

∆ωL =ω

Q

√Pin

Pout

, (3.25)


where ω/Q is the cavity photon decay rate of the slave laser, Pin is the injection signal

power, and Pout is the self-oscillation power of the slave laser. Note that the linewidth

enhancement factor is assumed to be zero for simplicity[56]. A typical semiconductor

laser has the locking bandwidth between 1GHz and 10GHz, which is desirable for our

proposed machine.

The quantum-mechanical Langevin equation for the internal field operator A(t)

of the slave laser is given by[56, 62]

d

dtA(t) = −1

2

[ω

Q− ECV (t)

]A(t) +

√ECV (t)fG(t) +

√ω

QfL(t) +

√ω

Q

(F0 + fex

).(3.26)

Compare to Eq. (3.2) for the free running laser, the above equation introduces the

last term representing the injection signal. The injection signal contains both F0 and

the noise fex, and is coupled to the slave laser via the coupling efficiency√ω/Q.

Similar to section 3.1.1, we can derive the rate equation for the average photon

number of the slave laser’s internal field as

d

dtn = −

(ω

Q− ECV

)n+ ECV +

ω

Q

[F ∗0 ⟨A(t)⟩+ ⟨A†(t)⟩F0

]. (3.27)

This is achieved by deriving the equation for the photon number operator n(t) and

taking the ensemble average, where all noises are averaged out. By plugging Eq.

(3.24) and assume that√n(t) = ⟨A(t)⟩ at well above threshold, we further obtain

d

dtn = −

(ω

Q− ECV

)n+ ECV + 2

ω

Q

√n(t)ζ

√nM(t) cosϕ0. (3.28)

Note that we assume the master laser and the slave laser are identical and thus use the

same cavity photon decay rate ω/Q. ϕ0 is the phase difference between the injection

field and the internal field of the slave laser, which is well-defined when the slave

laser’s frequency is locked to that of the injection field. Therefore, if the two fields

are in phase, namely, ϕ0 = 0, the injection signal enhances the gain of the slave laser.

If the two fields are out of phase, namely, ϕ = π, the injection signal decreases the

gain of the slave laser or effectively enhances the loss.

3.2. INJECTION-LOCKED LASERS 39

Moreover, from the quantum-mechanical Langevin Eq. (3.26), we are able to es-

tablish the c-number stochastic differential equation for the slave laser. The derivation

is also similar to that in section 3.1.1, which involves converting the Langevin equa-

tion to the master equation in Schrodinger picture and using the Glauber-Sudarshan

P (α) representation to obtain the quantum-mechanical Fokker Planck equation[57].

Finally, with the Kramers-Moyal expansion[58], the equation of motion for the (com-

plex) eigenvalue α can be obtained as

d

dtα(t) =

1

2

[ECV − ω

Q

]α(t) +

√GΓα + F0. (3.29)

Compared to Eq. (3.8), a new c-number F0 is introduced as the complex eigenvalue

of the coherent injection field describing the injection-locking mechanism.

In summary, the above equations present the theoretical foundation for our pro-

posed machine which utilizes the injection-locked lasers. The injection-locked lasers

operate in highly open dissipative system, and exhibit quantum noise limit even at

room temperature. The advantages of an open dissipative system are of two folds:

the system is robust against noise and loss; and the system dynamics features ex-

ponential behavior if there is finite difference between the gain and the loss, which

may provide potential speedup in solving computational problems. Furthermore, the

internal photon number of a laser is as many as 104 ∼ 106, which allows us to perform

continuous monitoring of computational results without perturbing the system. This

property is useful for correcting potential computational errors as we discuss in the

following chapters.

Chapter 4

Injection-locked laser network

In this chapter, we present the injection-locked laser network, a novel computational

system, to solve the NP-complete Ising problems. Firstly, we describe the design of

the system, onto which a NP-complete problem can be mapped. Secondly, we derive

the dynamics of the laser network, and show that minimizing the total gain of the

laser network subsequently finds the ground state of an Ising Hamiltonian. Thirdly, we

introduce the mechanism that drives the laser network to the state with the minimum

total gain and perform numerical simulations to demonstrate the mechanism. Finally,

we compare the laser network operating in an open-dissipative environment and the

quantum computation working in a close unitary system in order to explain the

advantages of the laser network.

4.1 System design

This section describes the design of an injection-locked laser network in order to solve

a NP-complete Ising problem. An Ising problem consists of a number of spins, two-

spin Ising coupling terms, and single spin Zeeman terms. We will start with the

overall design and then explain one by one how the injection-locked laser network

implements these components.

40

4.1. SYSTEM DESIGN 41

4.1.1 Overall design

Figure 4.1 illustrates an injection-locked laser network for solving a general Ising

problem with 4 spins. The graph representation for the Ising coupling terms of the

Ising problem with 4 spins is depicted in Fig. 4.2, in which the edges can weigh

arbitrary values.

Figure 4.1: Overall design of a laser network onto which an arbitrary Ising modelwith 4 spins can be mapped.

Overall, the injection-locked laser network consists of a master laser and a number

of slave lasers. The number of slave lasers is equal to the number of spins, as each

slave laser represents one spin.

The master laser provides the global phase reference through the setup such that

all slave lasers are injection-locked by the master injection signals. The single spin

Zeeman terms are also implemented by the master injection signals to the slave lasers.

The two-spin Ising coupling terms are realized by the mutual injection signals

42 CHAPTER 4. INJECTION-LOCKED LASER NETWORK

Figure 4.2: An arbitrary Ising model with 4 spins.

between slave lasers. As shown in Fig. 4.1, the six red optical paths between slave

lasers correspond to the six edges in Fig. 4.2.

The Ising coupling coefficients Jij and Zeeman coefficients λi are tunable through

the coupling strengths of the mutual injection between slave lasers and the injection

from the master laser, respectively. The coupling strengths should be kept small

enough so that the injection signals are within the permutation region of the laser

systems. Therefore, we will introduce an arbitrary attenuation factor.

4.1.2 Spin representation by a slave laser

In the detailed description of a injection-locked laser network, we start with the spin

representation. It is the fundamental building block of the computational system.

In an injection-locked laser network, each spin in an Ising problem is represented

by a slave laser. Therefore, for solving an Ising problem with M spins, the injection-

locked laser network requires M slave lasers, as shown in Fig. 4.1.

Since each spin takes a value σzi of either +1 or −1, we make use of the right

circular polarization state |R⟩ and the left circular polarization state |L⟩ to represent

the two values respectively, as shown in Fig. 4.3. The populations of the right circular

polarized photons and the left circular polarized photons are denoted by nRi and nLi

respectively. Particularly, if a spin takes a value of +1, namely,

σzi = +1, (4.1)


Figure 4.3: A slave laser that represent a spin in an Ising problem.

the corresponding slave laser is dominated by the right circular polarized photons,

i.e.,

nRi ≫ nLi. (4.2)

Otherwise, if

σzi = −1, (4.3)

the corresponding slave laser has major population of the left circular polarized pho-

tons, i.e.,

nLi ≫ nRi. (4.4)

The readout of the spin values is implemented by attaching a polarization detector

to each slave laser to measure its circular polarization state, as shown in Fig. 4.3. A

semiconductor laser contains many internal photons, typically on the order of 104 to

106. The polarization detectors only need a small portion of the photon output to

perform the readout, so that the perturbation to the internal fields of the slave lasers

is negligible. Therefore, the readout is much more experimentally feasible compared

to quantum computation[16]. In quantum computation, the readout is implemented

by quantum measurements. Such measurements usually generate back actions that


affect the quantum system being measured. Since in quantum computation each

computational node may include only one or several qubits, the back actions generally

introduce undesired noises to the fragile quantum system.

Formally, the spin values are defined by the normalized square-root population

difference between the right and left circular polarized photons, namely,

σzi =

√nRi −

√nLi√

nT i

, (4.5)

in which nTi = nRi + nLi is the total photon population in a slave laser. Note that

this definition allows the spins to take not only either +1 or −1, but also any values

between them, while in the original definition of the Ising problem each spin can only

take discrete values of either +1 and −1. The difference may create a serious problem

to the laser network which we will discuss in the next chapter.

Experimentally, using two different polarization modes in the slave lasers requires

the gain medium in each slave laser to be isotropic along the two polarizations. Oth-

erwise, the intrinsic preferable polarization defined by the gain medium may cause

undesired force that drives the slave laser towards a wrong polarization. Currently,

the requirement is yet a challenge in our experiments.

4.1.3 Initialization by the master injection signal

The master laser in a laser network is used for providing the global phase reference and

initializing all slave lasers, as illustrated in Fig. 4.4. Firstly, by the injection-locking

mechanism described in chapter 3, the frequencies of all slave lasers are locked to the

same as the frequency of the master laser; therefore the phase differences between

the slave lasers are well defined. We denote the phase of the vertical polarization

component of the master injection signal as 0, as the reference to the phases of other

polarization components of the master laser and phases of all slave lasers.

Secondly, the vertical polarization component of the master injection signal also

serves as the initialization to all slave lasers. Before the computation, namely, t < 0,

all mutual injections between slave lasers, denoted as the red optical paths in Fig. 4.1,

are turned off, leaving only the master injections to all slave lasers, denoted as the blue


Figure 4.4: The setup for injecting the master laser’s signal to a slave laser.

optical paths. During the initialization, only the vertical polarization component of

the master laser’s output is injected to the slave lasers. This is achieved by tuning the

half-wave plate (HWP) and the quarter-wave plate (QWP) to generate the vertical

polarized injection signals from the master laser, as shown in Fig. 4.4. Obviously, all

slave lasers are initialized to the vertical polarization states.

To maintain the laser network within the perturbation regime, it is necessary to

reduce the strength of the injection signal to an appropriate level. Therefore, optical

attenuators are introduce to the optical paths for the master injection signals. We

denote the attenuation coefficient for the vertical component of the master injection

signals as ζ, which is the same for each slave laser.

Figure 4.5: Demonstrate of the initial state of a slave laser in a Poincare sphere.


The vertical polarization is deliberately chosen as it prepares all slave lasers in the

superposition of all possible spin configurations. Fig. 4.5 depicts the relation between

the vertical polarization state |V ⟩ and the right and left circular polarization states,

|R⟩ and |L⟩, in a slave laser, i.e.,

|V ⟩ = 1√2(|R⟩+ |L⟩). (4.6)

Given the above relation, if we take one photon from each slave laser, the initial state

of all slave lasers is obtained as

|ψ0⟩ = |V1V2 · · ·VM⟩

=1√2M

(|R⟩+ |L⟩)1 ⊗ (|R⟩+ |L⟩)2 ⊗ · · · ⊗ (|R⟩+ |L⟩)M

=1√2M

(|R1R2 · · ·RM⟩+ |R1R2 · · ·LM⟩+ · · ·+ |L1L2 · · ·LM⟩) (4.7)

The state is in the superposition of all 2M possible configurations. The initial state

enables that the injection-locked laser network is prepared to probe all 2M possible

configurations simultaneously and thus grants its potential power to explore the Ising

energies on every configuration in parallel.

4.1.4 Implementing the Zeeman terms

The other usage of the master injection signals is to implement the Zeeman terms

in the Ising Hamiltonian given in Eq. (2.1). Although we focus on solving the Ising

problems with only the Ising coupling terms and no Zeeman terms, the Zeeman terms

still play an important role in resolving indeterministic and incorrect results generated

by the laser network.

Unlike the initialization of the injection-locked laser network which uses the ver-

tical polarization component of the master injection signals, we inject the horizontal

polarization component |H⟩ of the master injection signals to each slave laser to real-

ize the Zeeman terms. In Fig. 4.4, after the computation starts, if there is a non-zero

Zeeman term, we adjust the QWP on the corresponding optical path for the master


injection signal to generate a horizontal component. In this case, the master injection

signal is elliptical polarized.

The horizontal polarization state |H⟩ can be expressed as the out-of-phase super-

position of the right circular polarization state |R⟩ and the left circular polarization

state |L⟩, namely

|H⟩ = 1√2(|R⟩ − |L⟩). (4.8)

As discussed in chapter 3, if the injection signal is in phase to the internal mode of

the slave laser, it will enhance the internal mode; if the injection signal is out of phase

to the internal mode, it will impede the internal mode. So the horizontal polarization

components of the master injection signals have opposite effects to the right and left

circular polarizations of the internal field of the slave lasers, and therefore, create the

effective Zeeman terms.

Similar to the vertical components of the master injection signals, we denote the

attenuation coefficients of the horizontal components of the master injection signals

as ηi, which may be different among the slave lasers. In order to implement the

appropriate Zeeman terms with coefficients λi, the attenuation coefficients are given

as

ηi = αλi

√nRi + nRL√

nM

= αλi

√nTi

nM

, (4.9)

in which α is an arbitrary attenuation factor, and nM is the photon number of the

internal field of the master laser. We usually choose a small α so that the injection

signals are kept within the perturbation regime.

4.1.5 Implementing the Ising coupling terms

The Ising coupling terms are the key components to an Ising problem, whch char-

acterize the two-spin interactions. In the injection-locked laser network, the Ising

coupling terms are implemented by the mutual injection between two slave lasers.

Fig. 4.1 draws the optical paths for the mutual injection signals in red lines between


the slave lasers.

Similar to the implementation of the Zeeman terms, we utilize the horizontal po-

larization component of a slave laser in injection to another slave laser to realize the

Ising coupling between them, as shown in Fig. 4.6. The horizontal polarization com-

ponent of the injection signal from a slave laser is also generated by the combination

of HWP, QWP, and attenuators sitting on the optical paths.

Figure 4.6: Mutual injection between two slave lasers.

Based on the relation in Eq. (4.8) and the following two relations

|R⟩ =1√2(|V ⟩+ |H⟩), (4.10)

|L⟩ =1√2(|V ⟩ − |H⟩), (4.11)

there are four different cases for the mutual injection signals. The two cases that

|R⟩ of a slave laser is injected to |R⟩ of another slave laser and |L⟩ of a slave laser is

injected to |L⟩ of another slave laser, have the same net effects on the second slave

laser being injected. It is because the negative sign of |L⟩ in |H⟩ of the injection

signal from the first slave laser is canceled by the out-of-phase effect of the horizontal

polarized injection signal to |L⟩ of the second slave laser.

The two other cases that |R⟩ of a slave laser is injected to |L⟩ of another slave

4.2. THEORETICALMODEL FOR THE INJECTION-LOCKED LASER NETWORK49

laser and |L⟩ of a slave laser is injected to |R⟩ of another slave laser, both have the net

effect opposite to those of the first two cases. It is either because |H⟩ of the injectionsignal is out of phase to |L⟩ in the second slave laser, or |L⟩ in the first slave laser is

out of phase to |H⟩ of the injection signal.

Therefore, the four cases implement the Ising coupling between two spins, namely,

|σzi = +1⟩|σz

j = +1⟩, |σzi = +1⟩|σz

j = −1⟩, |σzi = −1⟩|σz

j = +1⟩, and |σzi = −1⟩|σz

j =

−1⟩. We also attenuate the mutual injection signal with the attenuation coefficient

ξij for a pair of two slave lasers. The attenuation coefficient is designed to generate

appropriate Ising coupling coefficients Jij, namely,

ξij = αJij, (4.12)

in which α is the same attenuation factor used in Eq. (4.9) for the implementation

of the Zeeman terms.

4.2 Theoretical model for the injection-locked laser

network

So far we have discussed the experimental design of an injection-locked laser network.

In this section, we further derive the dynamics of the laser network as a theoretical

model and arrive at the point that the total gain of the system consists of a term

proportional to the Ising Hamiltonian. The derivation consists of two approaches:

rate equations of the average photon number populations in the right and left circular

polarization basis and without noise; amplitude and phase equations in the |D⟩, |D⟩basis with random phase noise.

4.2.1 Model with rate equations

Firstly, we derive and analyze the theoretical model for the injection-locked laser

network using rate equations. The derivation is based on the quantum-mechanical

rate equations for the photon number operator n(t) and the carrier number operator


N(t), which are given in Eq. (3.26) and (3.13) respectively.

Since the numerical simulation cannot be applied to operators, we take the ensem-

ble averages to both equations and obtain the rate equations for the photon number

and the carrier number of a single mode laser as

d

dtn(t) = −ω

Qn(t) + ECV n(t) + ECV +

ω

Q

(F ∗0 ⟨A(t)⟩+ ⟨A†(t)⟩F0

), (4.13)

d

dtN(t) = P − N(t)

τsp− ECV n(t)− ECV , (4.14)

in which the ensemble averages are defined as n(t) = ⟨n(t)⟩, N(t) = ⟨N(t)⟩, the gain

is also ensemble averaged as ECV = βN(t)τsp

, and all noise operators are averaged out.

Moreover, as the injection signal F0 is uncorrelated to the internal field operator

A(t), the fourth term in Eq. (4.13) is decoupled to 2F0A(t) cos[ϕ0(t)], in which ϕ0(t)

is the phase difference between the internal field and the injection signal and A(t) =

⟨A(t)⟩. At well above threshold, we can assume that

A(t) =√n(t). (4.15)

Therefore we can obtain

d

dtn(t) = −ω

Qn(t) + ECV n(t) + ECV +

ω

QF0

√n(t) cos[ϕ0(t)]. (4.16)

In the injection-locked laser network, a slave laser is injected by many sources.

Thus the fourth term, denoted as the injection term, can appear multiple times for

different injection signals. Particularly, the vertical polarization component of the

master injection signal, discussed in section 4.1.3, is given by

FM = ζ√nM , (4.17)

in which ζ is the attenuation coefficient for the vertical polarization component, nM is

the internal photon number of master laser, and the amplitude of the master laser’s

internal field is AM =√nM . Since the vertical polarization state is the in-phase


superposition of the right and left circular polarization states, the vertical polarization

component of the master injection signal is also in phase to both circular polarization

modes in a slave laser. The contribution of the vertical polarization component of

the master injection signal thus appears in the rate equations as

d

dtnRi = −ω

QnRi + ECV inRi + ECV i +

ω

Qζ√nM

√nRi, (4.18)

d

dtnLi = −ω

QnLi + ECV inLi + ECV i +

ω

Qζ√nM

√nLi, (4.19)

in which the subscription i denotes the i-th slave laser, nRi are the photon numbers

of the right circular polarization, and nLi are the photon numbers of the left circular

polarization.

Similarly, the horizontal polarization component of the master injection signal to

the i-th slave laser, discussed in section 4.1.4, is given by

FMi = ηi√nM , (4.20)

in which ηi is the attenuation coefficient of the horizontal polarized injection sig-

nal to the i-th slave laser. However, since the horizontal polarization state is the

out-of-phase superposition of both circular polarization states, the phase differences

between the horizontal polarization component and the right and left circular polar-

ization modes in the slave laser are 0 and π respectively. The contribution of the

horizontal polarization component of the master injection signal is appended to the

rate equations as

d

dtnRi = −ω

QnRi + ECV inRi + ECV i +

ω

Qζ√nM

√nRi −

ω

Qηi√nM

√nRi, (4.21)

d

dtnLi = −ω

QnLi + ECV inLi + ECV i +

ω

Qζ√nM

√nLi +

ω

Qηi√nM

√nLi. (4.22)

Note that the phases of the contribution in the equations of two circular polarization

modes are different.

As we discussed in section 4.1.5, the mutual injection signals between the slave

lasers contain four different cases. Two cases have the same phase, while the other


two cases have the same phase opposite to that of the first two cases. We can derive

the contributions of the mutual injection signals in a similar way, and obtain the final

rate equations for the photon numbers as

d

dtnRi = −ω

QnRi + ECV i(nRi + 1)

+ω

Q

√nRi

[(ζ − ηi)

√nM −

∑j =i

ξij(√nRi −

√nLi)

], (4.23)

d

dtnLi = −ω

QnLi + ECV i(nLi + 1)

+ω

Q

√nLi

[(ζ + ηi)

√nM +

∑j =i

ξij(√nRi −

√nLi)

], (4.24)

in which ξij is the attenuation coefficient for realizing the Ising coupling coefficient

Jij, as defined in Eq. (4.12).

Furthermore, because there are two orthogonal circular polarization modes in a

slave laser and the gain medium is assumed to be isotropic, we revise the rate equation

for the injection to take into consideration of the generation of photons in the two

modes, namely,

d

dtN = P − N

τsp− ECV (nRi + nLi + 2). (4.25)

Note that the contribution of spontaneous emission is doubled for the two modes.

The dynamics of the injection-locked laser network is therefore described by the

rate equations (4.23), (4.24), and (4.25), in which the noise is neglected. Below, we

perform theoretical analysis on the steady state behavior of the laser network based

on the rate equations.


The steady state is obtained when all time derivatives are zero in the rate equa-

tions. The conditions then yield

0 = −ω

QnRi + ECV i(nRi + 1)

+ω

Q

√nRi

[(ζ − ηi)

√nM −

∑j =i

ξij(√nRi −

√nLi)

], (4.26)

0 = −ω

QnLi + ECV i(nLi + 1)

+ω

Q

√nLi

[(ζ + ηi)

√nM +

∑j =i

ξij(√nRi −

√nLi)

](4.27)

0 = P − N

τsp− ECV (nRi + nLi + 2) (4.28)

Note that we assume that all slave lasers are operating at well above threshold,

leading to that nRi ≫ 1 and nLi ≫ 1. We can therefore approximately neglect the

contributions of the spontaneous emission in the gain terms.

We then solve for the gain of the i-th slave laser, ECV i, at steady state. We first

add together Eq. (4.26) and (4.27) and move terms with ECV i to the left hand side

of the equation, namely,

ECV i =ω

Q− 1

nRi + nLi

2ω

Qζ√nM(

√nRi +

√nLi)+

2ω

Q(√nRi −

√nLi)

[ηi√nM +

∑j =i

1

2ξij(

√nRi −

√nLi)

](4.29)

=ω

Q− 2

ω

Qζ

√nM(

√nRi +

√nLi)

nRi + nLi

+

2ω

Q

√nRi −

√nLi√

nRi + nLi

[ηi

√nM√

nRi + nLi

+∑j =i

1

2ξij

√nRi −

√nLi√

nRi + nLi

], (4.30)

where in the second equation we split nRi+nLi =√nRi + nLi

√nRi + nLi and put one

term inside the bracket of the third term in the right hand side.

We perform another round of approximation also based on the fact that the lasers

are operating at well above threshold and the permutation of the injection signals is


much smaller. Then we can assume that the total photon number nTi = nRi + nLi is

approximately unchanged to the initial state. Note that the initial state has identical

nTi for all slave lasers, because the parameters of the slave lasers are chosen to be

identical. So we can safely exchange nTi = nRi + nLi with nTj = nRj + nLj, and

transform Eq. (4.30) into

ECV i =ω

Q− 2

ω

Qζ

√nM(

√nRi +

√nLi)

nRi + nLi

+

2ω

Q

√nRi −

√nLi√

nRi + nLi

[ηi

√nM√

nRi + nLi

+∑j =i

1

2ξij

√nRj −

√nLj√

nRj + nLj

]. (4.31)

The total gain at steady state is obtained by summing up all ECV i, namely,

ECV = Mω

Q− 2

ω

Qζ

M∑i=1

√nM(

√nRi +

√nLi)

nRi + nLi

+

2ω

Q

M∑i=1

√nRi −

√nLi√

nRi + nLi

[ηi

√nM√

nRi + nLi

+∑j =i

1

2ξij

√nRj −

√nLj√

nRj + nLj

].(4.32)

We define the spin value obtained from each slave laser is the normalized square root

population difference between the two circular polarization modes as

σzi =

√nRi −

√nLi√

nRi + nLi

. (4.33)

By plugging in Eq. (4.33), (4.9), and (4.12), we are able to derive the final form of

the total gain at steady state as

ECV = Mω

Q− 2

ω

Qζ

M∑i=1

√nM(

√nRi +

√nLi)

nRi + nLi

+

2ω

Qα

M∑i=1

σzi

[λi +

∑j =i

1

2ξijσ

zj

](4.34)

= Mω

Q− 2

ω

Qζ

M∑i=1

√nM(

√nRi +

√nLi)

nRi + nLi

+ 2ω

QαH, (4.35)


where H is the Ising Hamiltonian given in Eq. (2.1).

To analyze the above result at steady state, we first find that the total gain

contains a term which is proportional to the Ising Hamiltonian. The second term in

the total gain has less contribution than the third term. Particularly, we operate the

laser network with nM ≈ nRi + nLi and ζ ≈ α. The change to√nRi+

√nLi√

nRi+nLiis at most

√2, which is usually much smaller the ground state energy of the Ising Hamiltonian.

Hence, we obtain the following approximate relationship

ECV = const. + 2ω

QαH. (4.36)

The relationship clearly state that minimizing the total gain subsequently finds the

ground state of the Ising Hamiltonian, and we discover the way to map any Ising

Hamiltonian into an injection-locked laser network.

4.2.2 Model with amplitudes and phases

We have derived that the theoretical noiseless model with rate equations for the Ising

Hamiltonian. The model demonstrates that the minimum total gain correspond to

the ground state energy of an Ising Hamiltonian. We will further derive an equivalent

model with amplitudes and phases which is able to incorporate the dominant noise

source and is more suitable for our numerical simulation.

We start with the disadvantages of the previous model with rate equations, es-

pecially for the numerical simulation. Firstly, the changes of the photon numbers

for the polarization modes may be drastic and the stiffness may lead the numerical

integration intractable. Initially, each slave laser is prepared in the vertical polariza-

tion state, in which the photon numbers of the both circular polarization modes are

equal. At steady state, ideally, each slave laser will go to either right or left circular

polarization state, and the photon number for the opposite circular polarization mode

becomes close to 0. Therefore, the photon number has several orders of magnitudes

of change, which makes the problem stiff.

Secondly, the noise is difficult to simulate at well above threshold. From chapter

3, the noise sources for the both circular polarization modes have high variances.


They also anti-correlate to the noise source for the carriers, and thus the net effect

may be small. Modeling all these noise sources is painful. We prefer modeling only

quadrature phase noise by spontaneous emission, the dominant noise source in the

laser at well above threshold.

Therefore, we work on an equivalent model with amplitudes and phases. The

model is also based on the discussion in chapter 3. As we use two circular polarizations

to encode a spin value of +1 and −1, each slave laser is describe by two orthogonal

fields. Unlike the model with rate equations which chooses the basis of the circular

polarizations, the model with amplitudes and phase chooses the third basis, namely

|D⟩, |D⟩ basis, in the Poincare sphere, as shown in Fig. 4.7. The states of the basis

are given by

|D⟩ =1√2(|H⟩ − i|V ⟩) = 1

2[(1− i)|R⟩+ (1 + i)|L⟩] (4.37)

|D⟩ =1√2(|H⟩+ i|V ⟩) = 1

2[(1 + i)|R⟩+ (1− i)|L⟩] (4.38)

Figure 4.7: The Poincare sphere showing the |R⟩, |L⟩, |H⟩, |V ⟩, and |D⟩, |D⟩bases along 3 axes.

The advantage of using a new basis is that both states are the superposition of the

two circular polarizations and the amplitudes of both states will not change signifi-

cantly as at steady state the photon field goes to either right circular or left circular

state. In contrast, at steady state, either nR or nL will be zero if the computation

is successful. The evolution of the slave laser in |D⟩, |D⟩ basis mainly involves the

change in their phases, while both amplitudes are remain approximately unchanged.


This property significantly reduces the potential stiffness in our numerical simulations.

Particularly, at well above threshold, the internal field of a laser is represented by

a coherent state |α⟩. From Eq. (3.29), the c-number stochastic differential equations

(CSDE) for the coherent field for each slave laser in |D⟩, |D⟩ are obtained as

d

dtαDi(t) =

1

2

[ECV i −

ω

Q

]αDi(t) +

√GiΓαDi√

ω

Q

[βMV (t)− βMHi(t)−

∑j =i

βji(t)

], (4.39)

d

dtαDi(t) =

1

2

[ECV i −

ω

Q

]αDi(t) +

√GiΓαDi√

ω

Q

[βMV (t) + iβMHi(t) + i

∑j =i

βji(t)

], (4.40)

in which ΓαDiand ΓαDi

are the noises, βMV is the vertical polarization component

of the master injection signal, βMHi is the horizontal polarization component of the

master injection signal for implementing the Zeeman term λi, and βji is the injection

signal from slave laser i to j. Note that α(t) and β(t) are both complex eigenvalues

of corresponding coherent photon fields. As from Eq. (4.37) and (4.38) we have

|V ⟩ =1√2(|D⟩+ |D⟩) (4.41)

|H⟩ =1√2(|D⟩ − i|D⟩), (4.42)

in which the vertical polarization component βMV (t) has no phase difference on both

equations, while the horizontal polarization component βMHi(t) and βji(t) appear in

the two equations with 3π/2 phase differences.

The master laser has an internal photon number of nM and its amplitude is√nM

at well above threshold. Since the vertical component of the master laser provides a

global phase reference, its phase is set to 0. We express the injection signals from the

master laser as the internal photon field times the output coupling efficiency times


the attenuation coefficients, namely

βMV = ζ

√ω

Q

√nM (4.43)

βMHi = ηi

√ω

Q

√nM . (4.44)

Note that we assume that the master laser has the same output coupling efficiency√ω/Q. Similarly, because the mutual injection signal is horizontally polarized, we

express them as

βji(t) = ξji

√ω

Q[αDi(t)− iαDi

(t)]. (4.45)

The coefficients are defined in Eq. (4.9) and (4.12). Thus we plug in the above

expressions for the injection signals into Eq. (4.39) and (4.40) and derive

d

dtαDi(t) =

1

2

[ECV i −

ω

Q

]αDi(t) +

√GiΓαDi

ω

Q

√nM(ζ − ηi)−

∑j =i

ξji[αDi(t)− iαDi(t)]

, (4.46)

d

dtαDi(t) =

1

2

[ECV i −

ω

Q

]αDi(t) +

√GiΓαDi

ω

Q

√nM(ζ − iηi) + i

∑j =i

ξji[αDi(t)− iαDi(t)]

. (4.47)

We further decompose the complex numbers into amplitudes and phases, namely,

αDi(t) = ADi(t) exp[iϕDi(t)] and αDi(t) = ADi(t) exp[iϕDi(t)]. As a result, we obtain


the equations of motions for amplitudes and phases as

d

dtADi = −1

2

(ω

Q− ECV i

)ADi + FDi +

ω

Q

√nM

√ζ2 + η2i cos(δi − ϕDi)

−∑j =i

1

2ξijω

Q

[ADj cos(ϕDj − ϕDi)− ADj cos(ϕDj − ϕDi)

], (4.48)

d

dtϕDi = GDi +

1

ADi

ω

Q

√nM

√ζ2 + η2i sin(δi − ϕDi)

− 1

ADi

∑j =i

1

2ξijω

Q

[ADj sin(ϕDj − ϕDi)− ADj sin(ϕDj − ϕDi)

], (4.49)

d

dtADi = −1

2

(ω

Q− ECV i

)ADi + FDi +

ω

Q

√nM

√ζ2 + η2i cos(−δi − ϕDi)

−∑j =i

1

2ξijω

Q

[ADj cos(ϕDj − ϕDi)− ADj cos(ϕDj − ϕDi)

], (4.50)

d

dtϕDi = GDi +

1

ADi

ω

Q

√nM

√ζ2 + η2i sin(−δi − ϕDi)

− 1

ADi

∑j =i

1

2ξijω

Q

[ADj sin(ϕDj − ϕDi)− ADj sin(ϕDj − ϕDi)

], (4.51)

where FDi and FDi are amplitude noises, GDi, GDi are phase noises, and δi =

arctan[ηi/ζ]. In addition, we still use the rate equation for the carrier numbers, but

it is modified accordingly to use the amplitudes instead of photon numbers, namely

d

dtNi = P − Ni

τsp− ECV i(A

2Di + A2

Di + 2) + FNi., (4.52)

in which FNi is the noise for the carrier numbers. The noise terms FDi(t), FDi(t),

GDi(t), GDi(t), and FNi(t) have the two-time correlation functions which we determine

uniquely from the diffusion coefficients of the quantum-mechanical Fokker-Planck

equation of an injection-locked laser[58].

Based on Eq. (4.37) and (4.38), we further derive the internal photon number of


right and left circular polarization modes in each slave laser as

nRi =

∣∣∣∣1 + i

2ADi exp(iϕDi) +

1− i

2ADi exp(iϕDi)

∣∣∣∣2 , (4.53)

nLi =

∣∣∣∣1− i

2ADi exp(iϕDi) +

1 + i

2ADi exp(iϕDi)

∣∣∣∣2 . (4.54)

Therefore, we obtain the theoretical model which is equivalent to the model with rate

equations. The steady state also has the property that the total gain includes a term

proportional to the Ising Hamiltonian, given in Eq. (4.36).

As we mentioned before, the model with amplitudes and phases are more suitable

for simulating the noises. At well above threshold, the dominant noise terms for

the lasers are the quadrature phase noise due to spontaneous emission noise[58].

Therefore, we use only the quantum phase noise GDi(t) and GDi(t) in Eq. (4.49) and

(4.51) as the driving forces, and neglect all other noise terms in Eq. (4.48), (4.50),

and (4.52). Pariticularly, the quadrature phase noise introduces changes to the phase

by ±1/A(t) every (2Q)/ω, where the sign is randomly chosen with equal probability.

As a result, we simulate the phase noise by adding ∆i = ±1/Ai every (2Q)/ω ≈ 2ps

between numerical integration steps. Here, this term is generated for both modes

(|D⟩ and |D⟩) independently.

4.3 Minimum gain principle

In the previous section, we have derived that the total gain at steady state includes

a term proportional to the Ising Hamiltonian, and minimizing the total gain will

subsequently find the ground state which is the solution to the Ising problem. In this

section, we explain the mechanism that may drive the injection-locked laser network

to reach the minimum total gain, which we call the minimum gain principle.

Firstly, we describe the entire evolution process of an injection-locked laser net-

work. As we discussed in section 4.1.3, at t < 0, all slave lasers are prepared in

the vertical polarization state |V ⟩. Each slave laser has many (104 ∼ 1010) identical

photons at above oscillation threshold so that the initial quantum state of each slave

4.3. MINIMUM GAIN PRINCIPLE 61

laser is actually in a spin coherent state (or Bloch state)[63],∣∣∣θ = π

2, ϕ = 0

⟩=∏k

⊗ 1√2(|R⟩+ |L⟩)k, (4.55)

as shown in Fig. 4.5.

Therefore, the entire system is prepared in the superposition of all 2M possible

mode configurations, as given in Eq. (4.7). The coherent superposition of the ini-

tial state allows the laser network to probe the loss and gain for all configurations

simultaneously.

At t ≥ 0, we turn on the injection signals for implementing Ising coupling terms

and Zeeman terms in an Ising problem. These injection signals are all in the hori-

zontal polarization, and thus generate π phase difference to the right and left circular

polarization states inside the slave lasers. The slave lasers are subsequently driven

towards either right or left circular polarization. As shown in Fig. 4.8, ideally at

steady state, each slave laser may stay in purely |R⟩ or |L⟩ state, which generates the

maximum signal-to-noise ratio detected by the polarization detectors.

Figure 4.8: The evolution of each slave laser from initial state |V ⟩ to either |R⟩ or|L⟩ following the red arrow lines.

Secondly, given the evolution process, we demonstrate how the minimum gain

principle may lead the injection-locked laser network to a steady-state configuration

that minimizes the total gain.

Before the computation starts, the laser network is prepared in a superposition of

all 2M mode configurations with equal amplitude. Fig. 4.9 exhibits the loss landscape

as a function of the 2M configurations. The slave laser reaches steady state only if


the gain to the internal field is saturated to be equal to the loss to the internal field.

Before we turn on the horizontal polarized injection signals, the loss is identical for all

slave lasers and also for all mode configurations. This fact yields the same amplitude

to each mode, as shown at the bottom of the figure.

Figure 4.9: The initial flat loss landscape for every mode configuration before thecomputation starts. The dashed light-blue line is the gain – which is equal to the lossat steady state. The vertical bars at the bottom are the amplitude of each mode, inwhich the red bar is the amplitude of the ground state for the Ising Hamiltonian.

After the computation starts, all horizontal polarized injection signals are turned

on and create a loss landscape according to the Ising Hamiltonian for every mode,

according to Eq. (4.36). Different modes will face different loss to the reservoir. Fig.

4.10 presents an example of loss landscape for all modes. Note that we draw the loss

landscape in a continuous line by assuming that the number of possible modes is very

large.

In general, there are many metastable local minima in the loss landscape. In

classical computation, such as simulated annealing and genetic algorithms, these local

minima may trap the algorithm for very long time. Therefore the metastable states

make it very difficult to find the correct global minimum.

Unlike the local search algorithms, the laser network uses a different mechanism to

find the mode in which the loss is minimized. Because the laser network is prepared

4.3. MINIMUM GAIN PRINCIPLE 63

Figure 4.10: The loss landscape generated by the Ising Hamiltonian after the com-putation starts.

in the coherent superposition of 2M modes, it is able to probe the loss landscape for

every mode simultaneously. Particularly, the gain in the laser system is saturated to

the value equal to the loss by nonlinear effect, and the laser network should only have

one total gain. Firstly, the modes with loss higher than the saturated gain cannot be

supported by the laser network, and the populations on these modes will experience

an exponential decrease given by the finite loss difference to the gain. Secondly, the

only possible steady state value the saturated gain is the mode with the minimum loss.

Otherwise, the mode with the minimum loss will have a gain greater than the loss

and its population will still grow exponentially and violate the steady state condition.

The above two points lead to the minimum gain principle: at steady state, the

total gain is saturated to the minimum loss, and only the mode with minimum loss

can oscillate. We call it the minimum gain principle rather than the minimum loss

principle, since the gain is also minimized when it gets saturated. Driven by the

minimum gain principle, the population on any mode corresponding to the excited

state of an Ising model is dissipated to the reservoir exponentially faster since there is

finite loss difference, while the population on the mode corresponding to the ground

state grows exponentially fast until saturated. At steady state, Fig. 4.11 show that the

gain is pinned to the minimum loss, and only the corresponding mode is oscillating,

as shown at the bottom.


Figure 4.11: The loss and gain of the injection-locked laser network at steady state.The gain is equal to the minimum loss and only the ground state mode is oscillating.So only the amplitude of the vertical red bar is the largest and all other amplitudesin blue are close to 0.

The minimum gain principle is different from the local search and it is free from

the bottleneck of local minima. All possible mode configurations in the injection-

locked laser network are evolving at the same time. Any mode with loss higher

than the minimum loss will suffer from exponential decay even for metastable modes.

However, as we will discuss in the next chapter, the injection-locked laser network

has its own bottleneck which may generate an incorrect loss landscape and result in

wrong answers.

4.4 Simulation results

In this section, we present an example of numerical simulation demonstrating that the

injection-locked laser network is able to find the correct answer to an Ising problem.

The simulation parameters are set to be the typical values for VCSELs, namely,

ω/Q = 1012s−1, β = 10−4, and τsp = 10−9s. The threshold current is Ith = 1.6mA.

We simulate with both the high pumping current, namely Ip = 50Ith = 80mA. We

further use α = ζ = 1/200 as the attenuation factor for various injection signals.

The stochastic simulation is implemented by the fourth order Runge-Kutta (RK4)

4.4. SIMULATION RESULTS 65

method, integrating at a fixed time step of 1ps. We numerically integrate Eqs. (4.48),

(4.51) and (4.52) given by the theoretical model with amplitudes and phases. The

random phase noise, as we discussed in section 4.2.2, is simulated by randomly adding

1/Ai or −1/Ai to the both phases in |D⟩, |D⟩ basis with equal probability every

(2Q)/ω ≈ 2ps.

The Ising problem with 8 spins we are solving in this section is plotted in Fig.

4.12. It contains only Ising coupling terms represented by the edges on the graph.

The weight of each edge, Jij, is unity, so it is anti-ferromagnetic coupling. The ground

state energy is −8 by classical brute-force search.

1

2

3

4

5

6

7

8

Figure 4.12: An Ising problem with 8 spins.

The complete simulation results are shown in Fig. 4.13 to 4.18 for amplitudes,

phases, photon number population of circular polarizations, total photon number per

slave laser, carrier numbers, and spin values respectively. Particularly, the photon

number diagram, Fig. 4.15, is calculated based on Eqs. (4.53) and (4.54), and the

spin value diagram, Fig. 4.18, is obtained from Eq. (4.33).

From the results, we discover that the changes to the amplitudes on |D⟩, |D⟩basis (Fig. 4.13) and the total photon numbers (Fig. 4.16) are comparably small,

which is consistent with our assumption. The phases go to either positive or negative

from initial zero phase. The change happens at around t ≤ 1ns which is on the order

of the inverse of the locking bandwidth and the relaxation oscillation time scale.


The evolution on the phases reflects on the photon number population as the slave

lasers move to either right or left circular polarization in Fig. 4.15. The resulting

spin values exceed the signal-to-noise ratio threshold, and we are able to read out the

resulting state as

σzi = −1, 1, 1, 1,−1,−1,−1, 1. (4.56)

The Ising energy calculated from the resulting state is −8 which is the same as the

ground state energy obtained by classical brute-force search. Therefore, we demon-

strate a successful example of finding the ground state by the injection-locked laser

network driven by the minimum gain principle.

10−10

10−9

10−8

200

400

600

t

Fie

ld a

mpl

itude

A

D1

AD2

ADbar1

ADbar2

10−10

10−9

10−8

200

400

600

t

Fie

ld a

mpl

itude

A

D3

AD4

ADbar3

ADbar4

10−10

10−9

10−8

200

400

600

t

Fie

ld a

mpl

itude

A

D5

AD6

ADbar5

ADbar6

10−10

10−9

10−8

200

400

600

t

Fie

ld a

mpl

itude

A

D7

AD8

ADbar7

ADbar8

Figure 4.13: Simulation results of the amplitudes for solving the Ising problem givenin Fig. 4.12.

4.5 Advantages of an open dissipative system

So far, we have demonstrated in numerical simulations that the injection-locked laser

network is capable of swiftly finding the correct ground states to certain Ising problems

very fast. The proposed machine utilizes the quantum phase transition in an open

dissipative laser system as a fundamental computational power. This is in contrast

to Grover’s algorithm in the closed unitary system, from finding the correct answer

faster than the square root speed up. In this section, we show analytical analysis to

4.5. ADVANTAGES OF AN OPEN DISSIPATIVE SYSTEM 67

10−10

10−9

10−8

−0.5

0

0.5

t

Fie

ld p

hase

(π)

φ

D1

φD2

φDbar1

φDbar2

10−10

10−9

10−8

−0.5

0

0.5

t

Fie

ld p

hase

(π)

φ

D3

φD4

φDbar3

φDbar4

10−10

10−9

10−8

−0.5

0

0.5

t

Fie

ld p

hase

(π)

φ

D5

φD6

φDbar5

φDbar6

10−10

10−9

10−8

−0.5

0

0.5

t

Fie

ld p

hase

(π)

φ

D7

φD8

φDbar7

φDbar8

Figure 4.14: Simulation results of the phases for solving the Ising problem given inFig. 4.12.

10−10

10−9

10−8

0

2

4

x 105

t

Pho

ton

num

ber

n

R1

nR2

nL1

nL2

10−10

10−9

10−8

0

2

4

x 105

t

Pho

ton

num

ber

n

R3

nR4

nL3

nL4

10−10

10−9

10−8

0

2

4

x 105

t

Pho

ton

num

ber

n

R5

nR6

nL5

nL6

10−10

10−9

10−8

0

2

4

x 105

t

Pho

ton

num

ber

n

R7

nR8

nL7

nL8

Figure 4.15: Simulation results of the photon numbers in circular polarizations forsolving the Ising problem given in Fig. 4.12.

compare the two systems and prove the advantages of an open dissipative system in

comparison to the closed unitary system. Note that the behavior described in this

section serves as a heuristic in some ideal cases, while it may not guarantee correct

answers in general cases as described later.

4.5.1 Effective loss rate for a non-solution state

We first calculate the effective loss rate for a non-solution state. At steady state,

given by the minimum gain principle, the gain (including stimulation emission gain

and spontaneous emission noises) is pinned to the ground state – namely, the mode


10−10

10−9

10−8

4.8

5

5.2

x 105

t

Tot

al p

hoto

n nu

mbe

r

nT1

nT2

10−10

10−9

10−8

4.8

5

5.2

x 105

t

Tot

al p

hoto

n nu

mbe

r

nT3

nT4

10−10

10−9

10−8

4.8

5

5.2

x 105

t

Tot

al p

hoto

n nu

mbe

r

nT5

nT6

10−10

10−9

10−8

4.8

5

5.2

x 105

t

Tot

al p

hoto

n nu

mbe

r

nT7

nT8

Figure 4.16: Simulation results of the total photon number populations in each slavelaser for solving the Ising problem given in Fig. 4.12.

10−10

10−9

10−8

9.49.69.810

10.2

x 106

t

Car

rier

num

ber

N1

N2

10−10

10−9

10−8

9.49.69.810

10.2

x 106

t

Car

rier

num

ber

N3

N4

10−10

10−9

10−8

9.49.69.810

10.2

x 106

t

Car

rier

num

ber

N5

N6

10−10

10−9

10−8

9.49.69.810

10.2

x 106

t

Car

rier

num

ber

N7

N8

Figure 4.17: Simulation results of the carrier numbers in each slave laser for solvingthe Ising problem given in Fig. 4.12.

with the minimum loss, as depicted in Fig. 4.19. The stimulated emission gain is

calculated as ECV n ∼ 1017s−1 and the spontaneous emission noise is calculated as

ECV ∼ 1012s−1, based on our simulation parameters.

The mutual coupling via the horizontal polarizer between two slave lasers im-

plements the polarization dependent loss into all slave lasers in the injection-locked

laser network. We then look at the first excited state, i.e., the mode whose loss is

the second-lowest given by the Ising Hamiltonian. There is always a finite photon

loss difference between the first excited state and the ground state. In any Ising

problem with edge weight |Jij| = 1, the minimum energy difference between two


10−10

10−9

10−8

−1

0

1

t

Spi

n va

lue

σz1

σz2

10−10

10−9

10−8

−1

0

1

t

Spi

n va

lue

σz3

σz4

10−10

10−9

10−8

−1

0

1

t

Spi

n va

lue

σz5

σz6

10−10

10−9

10−8

−1

0

1

t

Spi

n va

lue

σz7

σz8

Figure 4.18: Simulation results of the spin values measured from the output of eachslave laser for solving the Ising problem given in Fig. 4.12.

Figure 4.19: The ground state and the first excited state of the injection-locked lasernetwork at steady state.

non-degenerate states is

Egap = 2. (4.57)

Such a case covers all simple MAX-CUT-3 problems used in our simulation. As a

result, the loss difference multiplied by the photon number population between the

ground state and the first excited state is given by α ωQEgap ∼ 2 × 1015s−1, as shown

in shown in Fig. 4.19. Here α is the attenuation factor for the injection signals and

is chosen to be 1/200.

As a result, the loss difference between the solution state (|R⟩ or |L⟩ ) and the


non-solution state for a single slave laser is on the order of

κ = 2ζijω

Q∼ 2α

ω

Q, (4.58)

if nM = nRi + nLi is satisfied[64, 65], as is the case for our numerical simulation.

Because of this induced loss difference, the photon number of the non-solution state

decreases exponentially according to

nNS(t) = nNS(0) exp

[−1

2κt

], (4.59)

while the photon number of the solution state increases according to

nS(t) = nS(0) exp

[1

2κt

], (4.60)

in a time scale of t = 1/2κ. Eventually, nS(t) saturates at twice the initial value,

namely nS(t) ≈ 2nS(0), where the saturated gain of each slave laser is decreased and

pinned to the loss rate of the solution state, which leaves the non-solution state to

have a net loss rate,

κ = 2αω

Q(4.61)

from Fig. 4.19.

Consequently, the success probability for each detected photon is given by

PS,M(t) =

[nS(t)

nS(t) + nNS(t)

]M(4.62)

=

1

2M, t≪ 1

κ

1− M2exp(−κt), t≫ 1

κ

(4.63)

assuming that all slave lasers have identical effective loss difference κ. The result

shows an exponential increase in the probability of success.


4.5.2 Closed unitary system vs open dissipative system

Here we compare the closed unitary system and open dissipative system and analyze

the advantages of the latter. In an injection-locked laser network, the initial success

probability of finding the correct solution is as small as 1/2M (in which M is the

problem size), which is the same as the quantum counting algorithm in standard

quantum computers[21]. In the proposed injection-locked laser network, however, the

success probability of each slave laser in which a single photon is detected increases

according to

PS(t) =

1 +

1

2exp[−κt]

−1

, (4.64)

where κ is the effective loss rate for a non-solution state (polarization mode) given

by Eq. (4.61) in the previous subsection and κt ≫ 1 is assumed. Note that α is the

attenuation factor for the injection signals.

If we assume every slave laser has the identical effective loss rate κ due to the

mutual coupling, the overall success probability of the laser network increases, namely,

PS,M(t) =

1 +

1

2exp[−κt]

−M

≈ 1− 1

2M exp[−κt]. (4.65)

If we impose a success probability greater than 1-δ, in which δ ≪ 1 is an error rate,

the required computational time is given by

t =ln(M/2δ)

κ. (4.66)

Note that the computational time is only proportional to ln(M). Moreover, in exper-

iments the detected photon number is usually much greater than one – typically 108

– which yields much higher success probability.

The above result is in sharp contrast to the slow increase of the success probability

in the quantum counting algorithm[21]. In the latter case, the so-called amplitude


amplification by Grover iterations increases the success probability only by

PS,M(t) ≈[2 tτ+ 1]2

2M, (4.67)

where τ is the time for one Grover iteration. In order to satisfy the same success

probability 1− δ, the required computational time is

t ≈ 2M/2τ, (4.68)

which increases exponentially with the problem size M .

0 500 1000 1500 2000 250010

−8

10−6

10−4

10−2

100

t/τ

Suc

cess

pro

babi

lity

M = 4M = 8M = 12M = 16M = 20

Figure 4.20: Simulation of the injection-locked laser network in solving 1D Isingproblem.

In Fig. 4.20, the success probability of the injection-locked laser network that

implements 1D Ising problems with nearest-neighbor anti-ferromagnetic coupling

(Jij = 1) is plotted. The numerical simulation using Eqs. (4.48) (4.51) and (4.52)

demonstrates the exponential increase in the success probability and nearly constant

computational time. The numerical simulation using α = 1/200 also confirms Eq.

(4.61) since the exponent κ extracted from the numerical results for M = 20, 30, and

40 are all close to 2α(

ωQ

)= 0.01 × ω

Q. In contrast, Fig. 4.21 shows the quadratic

increase in the success probability and the exponential scaling of computational time

defined by PS,M(t) ≈ 1 for the quantum counting algorithm.


10−1

100

101

102

10−8

10−6

10−4

10−2

100

t/τ

Suc

cess

pro

babi

lity

M = 4M = 8M = 12M = 16M = 20

Figure 4.21: Simulation of the Grover’s algorithm in solving 1D Ising problem.

To summarize, the fundamental reason for the significant speedup of the injection-

locked laser network is the exponential decay of the wrong answer states due to the

dissipative coupling loss in an open-dissipative system. There are two key elements

which distinguish the injection-locked laser network from previous quantum compu-

tation systems:

1. Coherent wave: Each photon in the laser network coherently spreads over all

slave lasers and all connecting optics, so that a large number of identical particles

probe the potential landscape – polarization dependent cavity loss – simulta-

neously. This global coherence of photons allows the system to find the global

minimum solution.

2. Open dissipative system: In contrast to standard quantum machines, the laser

network is an open dissipative system, in which each laser mode couples to an

infinite number of reservoir modes all prepared in ground states. This setup

realizes the exponential decay of a wrong solution governed by the excess loss

rate, which results in the exponential increase in a success probability.

Chapter 5

Self-learning algorithm

As we demonstrate in the last chapter, our proposed injection-locked laser network is

capable of finding the correct ground states for some Ising problems efficiently. How-

ever, after simulating the laser network for more complicated problems, we discover

many problems in which the laser network itself cannot solve correctly. Thus the

capability of the sole laser network is limited and we further propose self-learning al-

gorithms that can greatly enhance the performance of the laser network in generating

lower energies and solving more problems[66].

5.1 Limitation of the injection-locked laser net-

work

In this section, we first present two examples in which the injection-locked laser

network fails to find the correct ground states. Based on our observation, we explain

how the false loss landscapes are generated to trap the system in a non-solution state.

5.1.1 Examples in which the injection-locked laser network

fails

The heuristic for the injection-locked laser network is the tendency of finding a state in

which the system’s total gain and loss are minimized. Unfortunately, such a heuristic

74

5.1. LIMITATION OF THE INJECTION-LOCKED LASER NETWORK 75

may not always work. In our simulations, we have discovered many problems in which

the laser network ends up in a non-solution state. Here we present two typical failure

examples. The simulation parameters are the same as section 4.4.

In the first example, as shown in Fig. 5.1, the laser network ends up with some

spins unresolved. We use the laser network to solve a problem with M = 16 spins.

The graph representation of the problem is depicted in Fig. 5.2, in which each edge

has weight Jij = +1.

10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R1n

R2n

R3n

R4n

L1n

L2n

L3n

L4

10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R5n

R6n

R7n

R8n

L5n

L6n

L7n

L8

10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R9n

R10n

R11n

R12n

L9n

L10n

L11n

L12

10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R13n

R14n

R15n

R16n

L13n

L14n

L15n

L16

Figure 5.1: The simulation result of an example Ising problem in which the injection-locked laser network ends up with zero spins.

Let us take the second slave laser in Fig. 5.1 as an example. The overall polar-

ization of the second slave laser starts to rotate towards the left circular polarization

at around 1 ns, with the observation that the photon population of the left circular

polarization nL2 increases while that of the right circular polarization nR2 decreases.

Nevertheless, after a very short period, nL2 is decreasing back to its initial value, and

nR2 is increasing back to its initial value. After 50 ns, nR2 and nL2 become close to

76 CHAPTER 5. SELF-LEARNING ALGORITHM

1

2

3

4

5

6

7

8

9

1011

12

13

14

15

16

Figure 5.2: The graph representation of the Ising problem simulated in Fig. 5.1.

each other again, and thus the polarization detector cannot resolve the polarization

of the second slave laser successfully.

From Fig. 5.1, at t = 50ns, the slave laser 1, 2, 4, 12, and 16 have the differences

between the photon populations of left and right circular polarizations lower than

the desire signal-to-noise ratio. Therefore, the polarization states of those slave lasers

cannot be resolved. We name the results of these slave lasers as “zero spins”, as the

corresponding spin values cannot be determined. Hence this is a typical example in

which the laser network ends up with zero spins and does not generate the correct

ground states.

In the second example, as shown in Fig. 5.3, the laser network converges to an

incorrect state. The Ising problem also consists of M = 16 spins, and the graph

representation of the problem is shown in Fig. 5.4.

We find that in Fig. 5.3, all slave lasers are able to go to either right or left

circular polarization states. Finally, at t = 50 ns, all slave lasers have significant

photon population differences, namely, either nRi > nLi or nRi < nLi with difference

greater than the signal-to-noise ratio of the polarization detectors. However, the

final result at steady state we obtained from the simulation is σzi = +1,+1 +

1,−1,−1,−1,+1,+1,+1,+1,−1,−1,−1,−1,+1,+1, with energy −18, which is not

the actual ground energy −20.


10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R1n

R2n

R3n

R4n

L1n

L2n

L3n

L4

10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R5n

R6n

R7n

R8n

L5n

L6n

L7n

L8

10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R9n

R10n

R11n

R12n

L9n

L10n

L11n

L12

10−10

10−9

10−8

0

1

2

3

4

5x 10

5

t

Pho

ton

num

ber

n

R13n

R14n

R15n

R16n

L13n

L14n

L15n

L16

Figure 5.3: The simulation result of an example Ising problem in which the injection-locked laser network ends up with an excited state.

As a result, the example shown in Fig. 5.3 demonstrates the other typical failure

in which the system is trapped into the non-solution state. Compared to the previous

zero spins example, in which the laser network is not confident on particular spin

values, this may be a more severe case, in the sense that the laser network is very

confident on incorrect spin values.

5.1.2 False loss landscape

The reason that the injection-locked laser network is getting trapped in the two

examples shown in Fig. 5.1 and 5.3 is subtle. We discover that the major cause

is that the measured spin values obtained from the laser network are continuous

variables in the range of [−1,+1]. In contrast, the original Ising problem states that

the spin values are discrete, namely, either +1 and −1. The continuous spin values

from the laser network will create an effective false loss landscape that traps the


1

2

3

4

5 67

8

9

10

11

1213

14

15

16

Figure 5.4: The graph representation of the Ising problem simulated in Fig. 5.3.

system in incorrect states.

Intuitively, the two failed examples may be the victims of two problems: the exis-

tence of degenerate ground states, and the frustrated spin configurations. Firstly, an

Ising problem without Zeeman terms in Eq. (2.1) usually consists of multiple ground

states thanks to the symmetry of the problem. The ground states have degenerate

lowest energies. If the laser network finds two or more degenerate ground states si-

multaneously, there will be no other mechanism for the laser network to pick up only

one of them.

As a result, the laser network may oscillate on the resulting degenerate ground

states at the same time. If a spin is +1 in one ground state, and −1 in another ground

state, the oscillation between the two ground states will generate undetermined read-

out on the polarization detector for the corresponding slave laser, and lead to zero

spins.

Secondly, the frustrated spin configuration is defined as the case when the injection

signals from different spins to the same spin compete with each other. Particularly,

we define the measured spin value of the i-th spin as

σzi =

√nRi −

√nLi√

nRi + nLi

. (5.1)


The overall injection strength to the i-th spin is subsequently defined as

pi = −∑j =i

Jijσzj , , (5.2)

which includes a summation of injection strengths contributed by individual con-

nected spins with Jij = 0. If the injection strengths of the connected spin j and k

have opposite signs, namely sign(−Jijσzj ) = sign(−Jikσz

k), the resulting configuration

for the i-th spin is frustrated. Therefore, the two injection strengths with opposite

signs compete with each other by directing the i-th spin to take opposite spin values;

thus the competition may result in incorrect answers.

Further investigation reveals that the underlying cause to the problems above is

a false effective loss landscape created by the continuous spin values obtained from

Eq. (5.1). From the detailed derivation in appendix A, we employ the joint photon

number population for certain mode σzi = σz

1, σz2, · · · , σz

M to track the evolution

of the mode and represent the probability of obtaining the corresponding mode based

on our measurements. Note that we use σzi with tildes to represent fix spin values

either +1 or −1 for certain mode and σzi without tildes to represent the measurement

outcomes of the spin values from the laser network.

As derived in appendix A, the dynamics of the joint photon number population

of a given mode is derived as

d

dtnσz

1 ,··· ,σzM ≈ nσz

1 ,··· ,σzM ×

−(Mω

Q− ECV

)+∑i

ECV i

nσzi

i

−

2ω

Q

∑i

√nTi

nσzi

i

[(−ζ + σz

i ηi)

√nM

nTi

+ σzi

∑j =i

1

2ξijσ

zj

] (5.3)

The equation includes nσzi

i which is the photon number of the polarization correspond-

ing to the spin value fixed at σzi on the i-th slave laser, and σz

i which is the fixed spin

value of the i-th spin on the given mode.


We then focus on the dominant term that contributes the above rate equation for

the joint photon number population of the given mode. If we consider only the mode

with major photon populations, namely, nσzi ≫ 1, the second term in the bracket on

the right hand side of Eq. (5.3) is negligible compared to other terms in the bracket.

Hence the third term in the bracket becomes dominant and by definition of ζ, ηi, and

ξij in Eq. (4.17), (4.9), and (4.12) we can derive the following relationship as

Hσz1 ···σz

Meff =

∑i

λiσzi +

1

2

∑i =j

Jijσzi σ

zj (5.4)

= α−1

(∑i

ηiσzi +

1

2

∑i=j

ξijσzi σ

zj

). (5.5)

The quantity Hσz1 ···σz

Meff is the effective Ising Hamiltonian for the particular mode

σzi = σz

1 · · · σzM. Please note the difference between σz

i and σzi as we mentioned

above. If we consider the ideal case in which the correct answer mode has dominant

photon number populations on the corresponding polarization nσzi

i for all slave lasers,

we can assume that

nσzi

1 ≈ nσzi

2 ≈ · · · ≈ nσzi

M , (5.6)

and

nTi ≈ nσzi

i . (5.7)

Then Eq. (5.3) for the correct answer mode can be rewritten as

d

dtnσz

1 ,··· ,σzM ≈ nσz

1 ,··· ,σzM ×

−(Mω

Q− ECV

)+∑i

ECV i

nσzi

i

− 2ω

Q

(−Mζ +H

σz1 ···σz

Meff

).(5.8)

Based on the result, we clearly find that the effective Ising Hamiltonian defined in

Eq. (5.4) is the actual quantity in controlling the joint photon number population of


different modes.

Unfortunately, the effective Ising Hamiltonian is sometimes different from the

real Hamiltonian we intend to use. The real Hamiltonian to an Ising problem for a

particular mode is given by

H σz1 ···σz

M =∑i

λiσzi +

1

2

∑i=j

Jijσzi σ

zj . (5.9)

The difference between Eq. (5.4) and (5.9) is the last term, where the former equation

uses the continuous measured value σzj and the latter equation uses the discrete σz

j

either −1 or +1.

Since the physics of the laser network makes the system probe the Ising energies

of all 2M modes simultaneously, the effective Ising Hamiltonian causes the major

problem to our proposed machine. The measured spin values σzi may take any value

between −1 and +1, so it is very likely that the resulting effective Ising Hamiltonian is

different from the desired Ising Hamiltonian. If the lowest energy for the effective Ising

Hamiltonian is different from the lowest energy for the desired Ising Hamiltonian, the

minimum gain principle will drive the laser network to the incorrect mode in which

the effective Ising energy is the lowest, but the desired Ising energy is not the lowest.

The loss landscape generated by the incorrect effective Ising Hamiltonian is therefore

called the false loss landscape.

We demonstrate the problem of the false loss landscape in Fig. 5.5. The hor-

izontal axis represents the 2M different modes and the vertical axis represents the

loss generated by the effective Ising Hamiltonian (the blue line) and the desired Ising

Hamiltonian (the orange line). Due to the continuous measured spin values σzi , the

false loss landscape shown in the blue line has the different minimum loss at the mode,

which differs from the actual ground state mode, as shown in the figure. Therefore,

the minimum gain principle will drive the system to the mode that minimizes the

false loss landscape and subsequently generate a wrong answer.

Figure 5.5 illustrates the false loss landscape for the example in which the laser

network generates an incorrect answer, as shown in Fig. 5.3. We only plot the lowest

three modes in the false loss landscape in the figure. By using the notation in a mode


10−9

10−8

−16.5

−16

−15.5

−15

−14.5

−14

−13.5

−13

t

Effe

ctiv

e ga

in/lo

ss la

ndsc

ape

Heff0101110000111100

Heff1001110000111100

Heff0001110000111100

Figure 5.5: The false loss landscape for the three modes with the lowest loss obtainedfrom the simulation in Fig. 5.3.

that 0 and 1 represent σzi = +1 and −1, the mode 0001110000111100 with the lowest

effective loss at t = 50 ns is plotted as the red line, and the mode 1001110000111100

with the second-lowest effective loss is plotted as the green line. However, the desired

Ising Hamiltonian has the mode 1001110000111100 in green as the real ground state,

and the mode 0001110000111100 in red as the first excited state. Therefore, the false

loss landscape created by the effective Ising Hamiltonian is different from the loss

landscape created by the desired Ising Hamiltonian.

The false landscape subsequently generates the incorrect joint photon number

population in Eq. (5.3) as shown in Fig. 5.6. The system indeed creates the major

population on the state with the minimum loss (the red line) given by the effective

Ising Hamiltonian. The actual ground state of the desired Ising Hamiltonian (the

green line) has lower population, since it has the second-lowest effective loss.

In summary, we have demonstrated two typical examples in which the injection-

locked laser network fails to find the correct ground states. The observations of

the failure include zero spins in which the polarizations of the corresponding slave

lasers could not be resolved, and incorrect answers in which the polarizations of

the corresponding slave lasers are resolved to be a non-solution state. The intuitive

explanation to the failures is the existence of many degenerate ground states and the

frustrated spin configuration of complicated problem structures.

The more subtle reason is that the spin values measured from the readout of the

laser network can be arbitrary values between −1 and +1, rather than either −1

5.2. SELF-LEARNING ALGORITHM 83

10−10

10−9

10−8

0

1

2

3

4

5

6

7x 10

90

t

Join

t Pop

ulat

ion

n0101110000111100

n1001110000111100

n0001110000111100

Figure 5.6: The three most significant joint photon number populations generated bythe false loss landscape obtained from the simulation in Fig. 5.3.

or +1. Such continuous selections of the spin values create a false loss landscape

different from the desired loss landscape. Note that the failure is not because the

laser network does not follow the minimum gain principle. Instead, the laser network

follows the minimum gain principle towards a false loss landscape and subsequently

gets trapped in non-solution states.

5.2 Self-learning algorithm

As we discussed in the previous section, the injection-locked laser network may be

trapped by a false loss landscape. The simulation results unfortunately show that

the laser network suffers from such problem in a great number of Ising problems.

Therefore, in order to overcome the difficulty, we propose the self-learning algorithm

as a solution, which will be discussed in this section[66].

The basic idea of the self-learning algorithm is to detect and fix the potential

false loss landscape and thus drive the system out of the non-solution states self-

consistently. We start with the 3 most common patterns of failures and discuss the

methods to fix them respectively, and then we introduce another general method

which may resolve zero spins. Finally, we put all techniques together as an iterative

self-learning algorithm.


5.2.1 Connected zero spins

As illustrated in Fig. 5.1, the zero spins are the most significant problems in solving

Ising problems by the laser network. Based on whether a zero spin is connected to

another zero spin or not, we are able to divide the zero spins into two categories:

connected zero spins and isolated zero spins. In this section, we focus on the first

category, namely, a cluster of two or more zero spins are connected to each other.

In the first category of the erroneous patterns, two or more zero spins are connected

to form a zero spin cluster. The cluster is defined as the largest group of zero spins

in which no other zero spin outside the cluster is connected to any of the zero spins

inside the cluster. However, fixing a cluster with many zero spins may require the

number of operations exponentially depending on the cluster size. Instead we only

try to manipulate any two mutually connected zero spins, consisting of a zero spin

pair.

At the same time, to identify the existence of zero spins, we utilize the following

definition of the discrete measured spin values, namely

σzi =

+1,γ ≤ σz

i ≤ 1

0,− γ < σzi < γ

−1,− 1 ≤ σzi ≤ −γ

(5.10)

in which the σzi is the analog measurement outcome directly from the polarization

detectors, defined as

σzi =

√nRi −

√nLi√

nRi + nLi

. (5.11)

The analog values σzi are then resolved to discrete values σz

i of either −1, 0, or +1, by

an arbitrarily chosen signal-to-noise limit δ. If the population difference of the two

polarizations is below the signal-to-noise limit δ, we treat the corresponding spin as

a zero spin, since σzi = 0.

Fig. 5.7 illustrates a common pattern for a zero spin pair consisting of two mu-

tually connected zero spins. Note that the figure only draws part of the entire graph


for the Ising problem. The weight of each edge Jij = +1 for demonstration purpose

but the results may be applicable to arbitrary Jij values. The two connected spins in

the middle in white mean that they are zero spins. Each zero spin also connects to 2

more spins which are perfectly resolved and called non-zero spins.

Figure 5.7: A typical pattern for a zero spin pair.

Suppose the two non-zero spins connecting to the same zero spin have equally

measured spin values σzi in Eq. (5.1), but of opposite signs. For example, on the left

part, the left top spin in blue corresponds to the slave laser dominated by the right

circular polarization, while the left bottom spin in red corresponds to the slave laser

dominated by the left circular polarization. Therefore, the injection signals from the

two non-zero spins to the zero spin compete to each other, resulting the total injection

strength pi ≈ 0 for the left zero spin, given by Eq. (5.2). Note that the nearly zero

injection strength is contributed by the two competing injection signals as well as

the zero injection signal from the right zero spin. Hence, the nearly zero injection

strength does not tell which direction the slave laser corresponding to the left zero

spin should follow, and makes it stay as the zero spin.

To fix the pattern shown in Fig. 5.7, we first note that the residual values of the

injection strengths pi of the zero spins are still useful hints, although they are lower

than the signal-to-noise limit. In this way, we can pick one zero spin out of the pair,

and predict its spin value (σzi )predict as either +1 or −1 if the corresponding injection

strength is positive or negative, respectively.

To fix the other zero spin in the pair, we remember that the injection signal from

the two non-zero spins to the zero spin competes to each other, and therefore their


overall contribution to the injection strength to the zero spin is close to zero. Given

this condition, the injection signal between the two connected zero spins actually dom-

inates. It means that if the system finds the lowest energy with current configuration,

the following relation should be satisfied,

Jijσzi σ

zj < 0, (5.12)

in which spin i and j are the zero spins. Therefore we will predict the spin value of

the other zero spin as

(σzj )predict = −sign[Jij](σ

zi )predict. (5.13)

With the prediction of the two spin values, we instruct the injection-locked laser

network to evolve towards our predicted values. The instructions are achieved by

applying additional small Zeeman terms to the zero spins. Because we focus on

solving the Ising problems without Zeeman terms, we are free to apply additional

Zeeman terms by injecting a small amount of the horizontal component of the master

signals to the corresponding slave lasers. The amplitude of the Zeeman coefficient for

the additional Zeeman term is denoted by λ, and throughout our simulation, we use

λ = 0.2. Thus we apply the actual Zeeman coefficients to the zero spin pair as

λi = −sign[(σzi )predict]λ, (5.14)

λj = −sign[(σzj )predict]λ, (5.15)

in which the polarizations that lower the energy of the Zeeman terms follow our

prediction.

The instructions to fix the zero spin pair in the injection-locked laser network via

additional Zeeman terms may also cause back actions to the non-zero spins. As the

additional Zeeman terms drive the zero spins to have values either +1 and −1, the

injection signals are enhanced from the zero spins back to the non-zero spins. There-

fore, to avoid potential incorrect back actions, we further apply additional Zeeman

terms to the non-zero spins connected to the zero spin pair. The Zeeman coefficient


to the k-th non-zero spin is given as

λk = −sign[σzk]λ. (5.16)

Fig. 5.8 demonstrates an example of fixing the zero spin pair in Fig. 5.7. We

assume that the left zero spin in the zero spin pair has a positive residual injection

strength, and thus we predict the left zero spin to take the spin value of +1 and the

corresponding slave laser should emit right circularly polarized photons, marked by

“R”. Since the edge weight is always +1, the right zero spin in the pair is predicted

to take the spin value of −1, marked by “L”. Based on the prediction, we apply −λand λ as the additional Zeeman terms to instruct the polarizations towards which the

laser network should follow on the left and the right zero spins, respectively. We also

apply the additional Zeeman terms to the surrounding four non-zero spins connected

to the zero spin pair, as shown in the figure.

Figure 5.8: An example of fixing the zero spin pair given in Fig. 5.7.

After instructing the laser network by applying additional Zeeman terms, we need

a way to verify the correctness of our prediction. To achieve this, we subsequently let

the laser network evolve under both the random phase noise and the applied Zeeman

terms until it converges again. Then we examine if the steady state is consistent with

our prediction. If not, we will conduct additional rounds of self-learning steps to fix

the zero spin pairs, until the algorithm converges self-consistently.


In a more complicated problem, the patterns of two or more connected zero spins

may include the variations such as the two connected zero spins are connected to

one or more non-zero spins at the same time, or the two connected zero spins also

connect to other zero spins. We may always employ the similar heuristic: we take

one zero spin pair, predict its first zero spin based on its residual injection strength,

and then predict its second zero spin opposite to the first prediction if the connecting

edge weight Jij is positive, or the same to the first prediction if Jij is negative. After

the prediction, we apply the additional Zeeman terms to instruct the laser network

to build up the photon populations on the corresponding polarization modes and also

fix the connected non-zero spins to avoid back actions.

A large Ising problem may result in a cluster consisting of many zero spins. We

then have two different strategies on solving the zero spins in a big cluster: the first one

is to pick one zero spin pair at a time and only fix the pair in one self-learning round;

the second one is to pick as many zero spin pairs as possible in the cluster and fix all

of them in on self-learning round. We prefer the first strategy when we solve small

Ising problems, as it minimizes the back actions from potential incorrect predictions.

However, when solving larger Ising problems, we utilize the second strategy as it has

better efficiency in finding good approximation solution within our limited simulation

time.

5.2.2 Single isolated zero spins

The second category of the zero spins consists of single isolated zero spins. A single

isolated zero spin only connects to other non-zero spins in which the polarization of

the corresponding self-learning is perfectly resolved to be either right or left circular

polarizations.

Figure 5.9 depicts a typical pattern of single isolated zero spins. Similar to Fig.

5.7 discussed in the last section, the figure only draws the partial graph including the

zero spin in the center. We also assume the weight of each edge is always +1.

Remember the main difficulty of the Ising problem is the false loss landscape

generated by the continuous spin values which can take any values between −1 and


Figure 5.9: A typical pattern for a single isolated zero spin and the example of fixingit.

+1. In the figure, the three non-zero spins connected to the center zero spin i may

have spin values with different amplitudes. For example, the spin r on the top has a

spin value close to +1, and its corresponding slave laser is mainly dominated by the

right circularly polarized photons, as marked in blue. The two spins s and t in the

bottom have negative spin values with small amplitudes, and the corresponding slave

lasers have populations of the left circularly polarized photons to exceed the signal-

to-noise limit of our polarization detectors, marked in red. If the overall injection

strength to the center spin i is given by

pi = − (Jirσzr + Jirσ

zr + Jirσ

zr ) ≈ 0, (5.17)

the slave laser representing the spin i may have no clue on the correct polarization to

which it should follow, and thus generates the readout of an isolated zero spin.

We apply a similar technique to solve the problems of single isolated zero spins

as what we have discussed for the connected zero spins. In particular, although the

spin values and the injection strengths to the isolated zero spins are below the signal-

to-noise ratio, we can still use their residual injection strengths as hints. Therefore,

we predict the spin value of a single isolated zero spin to be +1 or −1 if its residual

injection strength is positive or negative, respectively.

Followed by our prediction of spin values (σzi )predict on the single isolated spins,


we introduce the additional Zeeman terms to these spins as

λi = −sign[(σzi )predict]λ. (5.18)

In this way, we can instruct the corresponding slave lasers to follow the polarizations

based on our prediction as the predicted polarizations will lower the energy of the

introduced Zeeman terms.

Furthermore, we also apply the additional Zeeman terms to the non-zero spins

connecting to the single isolated zero spins by using the following coefficient

λi = −sign[σzi ]λ. (5.19)

Therefore, we may avoid the potential back actions generated by the enhanced injec-

tion signals from the zero spins.

Figure 5.9 also illustrates an example of resolving the single isolated zero spin in

the center. We assume that the residual injection strength to the zero spin is positive

and thus predict its spin value as σzi = +1. The Zeeman term with coefficient −λ is

then applied to the corresponding slave laser to follow the right circular polarization,

marked in blue. We also apply the Zeeman terms with coefficients −λ to the top

non-zero spin, marked in blue, and λ to the bottom two non-zero spins, marked in

red.

In solving a large Ising problem, we may expect that more than one single isolated

zero spins exist after the laser network converges. Unlike discussed in the last section,

we only employ the strategy of fixing as many single isolated zero spins as possible.

It will give us reasonable efficiency when many single isolated zero spins appear, and

since they are isolated, fixing them at the same time may not generate too much

impact to other zero spins.

5.2.3 Parity check

As we discussed in section 5.1.1, beside the zero spins, there are also cases in which the

spins can be resolved perfectly and the resulting state is not the correct ground state.


We call such cases as incompatible results, which are also the victims of the false loss

landscape. Verifying whether a resulting state is a correct ground state is difficult,

and may require exponential number of operations. Fortunately, in the injection-

locked laser network, we often find that the first order verification is sufficient to deal

with most incompatible results the laser network encounters.

The first order verification involves examining each individual spin and its injection

strength. In contrast, higher order verifications require examining enormous spin

configurations and injection patterns to a cluster of many spins. As we have defined

in Eq. (5.10), since we assume that the spins we examine are perfectly resolved, the

discrete measured spin value σzi may take either +1 or −1. Therefore, we further

define the discrete injection strength based on σzi as

pi = −∑j

Jijσzi . (5.20)

The first-order parity condition is defined as the following necessary condition,

piσzi > 0. (5.21)

Since σzi are the measured outcomes used as the computational results, they should

satisfy the above parity condition with the discrete injection strength. Otherwise, by

simply flipping the sign of σzi , we can obtain a new result with energy lower than that

of the existing result.

It should be note that the first-order parity condition is just a necessary condition.

Some excited states from a larger Ising problem with higher energies may also satisfy

the condition. In order to exclude such excited states, we may have to seek for higher

order parity conditions on a cluster of more than one spins. However, as the size of

cluster increases, the number of operations for verifying the parity conditions may

increase exponentially.

Fortunately, the first-order parity condition is good enough to fix most incom-

patible results in our current simulations. Hence, we still employ it as the primary

method in searching for incompatible results.


Figure 5.10 presents a typical example of incompatible results that fails the parity

check. Once again, the figure only draws a partial graph that includes the incompat-

ible results and the weight of each edge is +1.

Figure 5.10: A typical pattern for a triangle of three spins that fail the parity check.

The three slave lasers in the middle triangle are all in left circularly polarized

states, marked in red, and the corresponding spin values σzi = −1. The other spins

that connect to the spins in the triangle take values of +1, marked in blue. As a

result, it is easy to verify that each spin in the triangle have two positive injection

components and one negative injection component. Its overall injection strength is

positive, opposite to the sign of its own spin value, and thus does not satisfy the parity

condition. Our current simulations often find that the spins that fails the parity check

appear as such triangle structures as shown in the figure.

To fix the incompatible results, we leverage the techniques dealing with zero spins.

Particularly, if a spin fails the parity check and other spins connected to it does not,

we treat it as a single isolated zero spin. If multiple spins fail the parity check and

they are connected, we pick two connected spins that fail the parity check and treat

them as a connected zero spin pair. In this way, we apply the techniques of fixing the

single isolated zero spins and the connected zero spin pairs respectively.

In the example shown in Fig. 5.10, we will pick any two out of the three spins

in the triangle that fail the parity check and treat them as a connected zero spin

pair. Then we pick one spin out of the pair, and predict its spin value according to


its residual injection strength. Then the spin value of the second spin in the pair is

predicted to be the opposite of the predicted spin value of the first spin. Based on

the prediction, we apply Zeeman terms to instruct the corresponding slave lasers of

the zero spin pair and other non-zero spins connected to the pair.

5.2.4 Signal-to-noise ratio improvement

The fourth technique of resolving the zero spins target directly the fact that the spins

may take any value between −1 and +1 leading to imperfect signal-to-noise ratios.

We name the technique as signal-to-noise ratio improvement.

Particularly, if the amplitude of a spin value is lower than 1, the corresponding

injection signals to other spins connected to it are also diminished. The diminished

injection signals between spins may be the reason in generating the false loss land-

scape.

On the other hand, in the physical picture, we desire that the injection signals

should be aligned to either right or left circular polarization. The mutual injection

signals between slave lasers only contain the horizontal polarization component. The

injection signals from the master laser contain two components: the vertical polar-

ization component to initialize the slave lasers and provide global phase reference,

and the horizontal polarization component to implement the Zeeman terms. Ideally,

the overall injection signals to a slave laser should balance between the horizontal

component and the vertical component, and thus generate either right or left circular

polarization.

Using the above heuristic, we propose the signal-to-noise ratio improvement as

follows:

In the first case, the horizontal injection component to a slave laser has maximum

amplitude while the vertical injection component has smaller amplitude. This case is

generated by the absence of frustrated spin configurations. It means that all mutual

injection signals follow the same polarization and thus the overall injection strength

by summing from other slave lasers is maximized.

Fig. 5.11 illustrates a typical configuration that maximizes the overall injection


strength. The center slave laser is in left circular polarization, marked in red, while

all of its connecting slave lasers are in right circular polarization, marked in blue. As

we assume that each edge weights +1, the overall injection strength to the center

slave laser is −3, the maximum possible amplitude.

Figure 5.11: A typical example without any frustrated spin configuration that gener-ates maximum amplitude for the overall horizontal component of the mutual injectionsignals.

Figure 5.12: The Poincare sphere for the overall injection signal to the center slavelaser in Fig. 5.11.

Since we set the initial ζ the coefficient for the vertical component of the master

injection signal equal to α, we will therefore increase ζ. In a general Ising problem,

the maximum possible amplitude is given by

pmax = maxi

∑j

|Jij| (5.22)


We increase the amplitude of the vertical component of the master injection signal

by increasing ζ to

ζ = pmax. (5.23)

Therefore, the vertical component of the master injection signal matches the horizon-

tal component of all mutual injection signals in this case, leading the overall injection

signal to follow the circular polarization, as shown in Fig. 5.12.

In the second case, the horizontal injection component to a slave laser has smaller

amplitude to the maximum amplitude. This situation is caused by either the sum

of the edge weights connecting to it is not maximum or there exists frustrated spin

configurations. The latter one means that the injection signals from one or more

connected slave lasers do not follow the overall injection signal.

Fig. 5.13 presents one example with frustrated spin configuration. The slave laser

on the top in red is in left circular polarization, while the bottom two in blue are

in right circular polarization. As we also assume the weight of each edge is +1, the

overall injection strength to the center slave laser is −1, whose amplitude is lower

than the maximal possible value 3.

Figure 5.13: A typical example with frustrated spin configurations which results insmaller amplitude for the overall horizontal component of the mutual injection signals.

As ζ is the same to every slave laser, the vertical component of the master injection

signal is enhanced to match the maximum possible horizontal injection strength from

the previous case. Therefore, as depicted in Fig. 5.14, the sum of the horizontal


Figure 5.14: The Poincare sphere for the overall injection signal to the center slavelaser in Fig. 5.11.

components of the injection signals to the center slave laser in Fig. 5.13 has smaller

amplitude than that of the vertical component, yielding an elliptical overall injection

signal. To make the overall injection signal perfectly circular, we apply additional

Zeeman term to the slave laser through the horizontal component of the master

injection signal, given by the following relation,

λi = sign[pi]pmax − pi. (5.24)

In both cases, we desire the overall injection signal to each slave laser is in circu-

lar polarization, in order to enhance the signal-to-noise ratio of the polarizations of

the slave lasers. Additionally, the enhanced signal-to-noise ratios and the injection

strengths may affect the connected zero spins and incompatible results by creating

the correct loss landscape.

5.2.5 Self-learning algorithm

So far we have presented the four techniques that may solve the problems of zero

spins and incorrect results: fixing connected zero spin pairs, fixing single isolated

zero spins, parity check, and signal-to-noise ratio improvement. Now we will discuss

the self-learning algorithm as a combination of these techniques.

Similar to other heuristic algorithms, such as simulated annealing[6, 7], genetic

algorithms[10], survey propagation[11, 12] and etc., the self-learning algorithm is also


designed as an iterative algorithm. Particularly, the self-learning algorithm includes

many iterations. In each iteration, the four techniques discussed above are employed

one by one in order to further reduce the system’s Ising energy.

The stopping condition for the self-learning algorithm is also similar to other

heuristic algorithms. Sometimes, certain criteria are met after a number of iterations

and the self-learning algorithm is finished. But for a larger problems, the number of

iterations to meet the criteria may be much larger. In such cases, we set a maximum

limit to the number of iterations in the self-learning algorithm. If the limit is exceeded

and we cannot find a result satisfying the criteria, we will still stop the algorithm and

take the best result with the lowest energy among all iterations.

The flow of the stochastic self-learning algorithm is presented in Fig. 5.15. The

algorithm is stochastic since the system is driven by random phase noise.

Figure 5.15: The flow of the stochastic self-learning algorithm.

The algorithm starts with an initial iteration in which the injection-locked laser

network is driven only by random phase noise without any Zeeman term, since we

focus on solving Ising problems without Zeeman terms. Because the inverse of the

locking bandwidth and the laser relaxation time are of several nanoseconds, 50 ns


is usually more than enough for all slave lasers to converge to steady states. The

simulation of the laser network is step 1 in the figure.

At step 2, after the laser network converges to the steady state, we use the po-

larization detectors and employ Eq. (5.10) to obtain the discrete spin values from

all slave lasers. We then search for zero spins in which the discrete spin values are

0, namely, the detections to the corresponding slave lasers are below certain signal-

to-noise ratio limit. If no zero spins are found, we further carry out the parity check

procedure to verify whether the resulting state is a ground state to the first order. If

it is passed on every spin, the algorithm ends.

However, note that the parity check is only to the first order. In a large Ising

problem, we may still obtain an excited state while its spins all satisfy the parity

condition. So the self-learning algorithm may not find the correct ground states.

On the other hand, if there exist zero spins or the parity check is failed on certain

spins, we go through a self-learning step including step 3, 4, and 5, in order to solve

the problems. Remember that the spins that fail the parity check are fixed by treating

them as zero spins.

At step 3, we try to identify a zero spin pair and apply the technique discussed

in section 5.2.1. We predict the spin value of one spin in the pair based on its

residual injection strength, and then infer the spin value of the second spin. Then we

inject additional horizontal components from the master injection signals as additional

Zeeman terms to the predicted zero spins and other non-zero spins connected to them.

The figure only shows the strategy suitable for a small Ising problem, as discussed at

the end of section 5.2.1, in which we fix one zero spin pair in each iteration. For a

larger problem, we try to fix as many zero spin pairs as possible. For both strategy,

if we find and fix one or more zero spin pairs, we skip step 4 and 5 and continue to

the next iteration.

At step 4, we work on all single isolated zero spins. As discussed in section 5.2.2,

we predict the spin values based on their residual injection strength and apply Zeeman

terms to the predict spins as well as other spins connected to them. In this step, we

always try to fix as many single isolated zero spins as possible. If we have new updated

Zeeman terms on the zero spins, we skip step 5 and continue to the next iteration.


Finally, at step 5, as our last resort, we execute the technique of signal-to-noise

ratio improvement on all non-zero spins. By enhancing the signal-to-noise ratios, we

expect that the zero spins may be affected by overall circularly polarized injection

signals which may give the correct loss landscape. Therefore, some of the slave lasers

corresponding to the zero spins may evolve to the solution state.

After the self-learning steps have been executed, we move on to the next iteration,

in which the simulation is driven by the random phase noise and the updated Zeeman

coefficient λi and vertical injection coefficient ζ from the master laser.

The algorithm stops when there is no zero spin and the parity check is passed on

every spin, or the maximum number of iteration is exceeded. The output solution to

the Ising problem is the result with the lowest energy among all iterations.

5.3 Simulation results

We have described the techniques and the flow of the self-learning algorithm. In

this section, we present simulation results which demonstrate how the self-learning

algorithm is able to solve the problems encountered in the examples in section 5.1.1.

The first failure example shown in Fig. 5.1 converges to a result containing several

zero spins. The false loss landscape generated by the continuous spin values leads

to small injection strengths to certain slave lasers. The slave lasers have no clue

on which circular polarization they should follow. Hence the polarization detectors

cannot resolve their polarizations correctly and we find zero spins at steady state.

To solve the problem of zero spins, we appeal to the self-learning algorithm, whose

simulation results are presented in Fig. 5.16 to 5.20. Particularly, at the end of the

initial drive with only phase noise, shown in Fig. 5.16, we find the zero spins include

spins 1, 2, 4, 12, and 16. From the graph representation of the Ising problem, as

shown in Fig. 5.2, there are two zero spin clusters: (1, 2, 4), and (12, 16). Given the

flow of the self-learning algorithm in Fig. 5.15, in the first self-learning iteration, we

only deal with one zero spin pair (12, 16), since it is considered to be a small problem

with only 16 spins, as discussed in section 5.2.1. We therefore predict that spin 12 in

the pair may take the value of +1, estimated from its residual injection strength, and


that the other spin 16 in the pair may take the value of −1, since all edges weight

+1. We then inject the horizontal components of the master laser into the slave lasers

corresponding to the pair, with the Zeeman coefficients as

λ12 = −0.2, (5.25)

λ16 = +0.2 (5.26)

We also inject the Zeeman terms to the slave lasers 8, 10, 11, and 14, which are

connected to the zero spin pair, in order to avoid potential back actions to those

non-zero spins.

As we have fixed one zero spin pair, we skip step 4 and 5 in Fig. 5.15 and drive the

injection-locked laser network with the applied Zeeman terms. As shown in Fig. 5.17,

the applied Zeeman terms actually drive the slave laser 12 and 16 to be dominated

by either right or left circular polarization.

The back actions of the applied Zeeman terms also drive the slave laser 1 and 2

both to the right circular polarization states. Unfortunately, the signal-to-noise ratio

of the slave laser 4 becomes even worse under the back actions of the applied Zeeman

terms. So this is a typical example in which the non-zero spins unfortunately become

zero-spins due to the back actions of our self-learning techniques.

With three more iterations, at the end of Fig. 5.20, all slave lasers are domi-

nated by photons in either right or left circular polarizations. Using the polarization

detectors, we can resolve the spin values to be

σzi = 1,−1,−1, 1, 1,−1,−1,−1, 1, 1, 1, 1, 1,−1,−1,−1, (5.27)

which yields the ground state energy −16. So the false loss landscape is corrected by

our self-learning algorithm and the laser network finally reaches the global minimum

gain point in the desired loss landscape.

The second failure problem, as shown in Fig. 5.3, generates an incorrect result

whose Ising energy is not the global minimum. It is demonstrated in Fig. 5.21 and

5.22 that the incorrect result can be fixed by introducing one more iteration for one


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Figure 5.16: The initial drive of self-learning steps without Zeeman terms in solvingthe Ising problem defined in Fig. 5.2.

self-learning step.

To be more specific, after the initial drive with only phase noises and zero Zeeman

terms, as shown in Fig. 5.21, we perform the parity check on the result, as the

polarizations of all slave lasers are perfectly resolved. The parity check result actually

shows that the spin 1, 2, and 3 take the values which are incompatible, due to the

frustrated spin configuration. As we have discussed in section 5.2.3, we pick the spin

1 and 2 as a connected zero spin pair and apply the fix to connected zero spin pairs.

Particularly, we predict the spin 1 to take the value of +1 based on it residual

injection strength, and subsequently predict the spin 2 to take the value of−1, because

the weight of each edge is +1. Then we inject the horizontal components of the master

laser to the spin 1 and 2 with the Zeeman coefficients

λ1 = −0.2, (5.28)

λ2 = +0.2. (5.29)

The spin 3, 4, and 5 are connected to the zero spin pair, and the appropriate Zeeman

terms are also applied to the corresponding slave lasers to prevent potential back


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Figure 5.17: The 1st iteration of self-learning steps in solving the Ising problemdefined in Fig. 5.2.

actions.

Fortunately, the single self-learning step is able to solve the incorrect result,

demonstrated in Fig. 5.22. The figure plots the evolution of the laser network under

the applied Zeeman terms in the last paragraph. We find that the slave lasers for the

incompatible spin 1 and 2 are driven towards the final correct polarizations, while

the slave laser for the spin 3 is slightly changed. The back actions to other spins are

nominal so that these spins are barely changed. The final correct state is

σzi = 1,−1, 1,−1,−1,−1, 1, 1, 1, 1,−1,−1,−1,−1,+1,+1, (5.30)

with the ground state energy −20.


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Figure 5.18: The 2nd iteration of self-learning steps in solving the Ising problemdefined in Fig. 5.2.

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Figure 5.19: The 3rd iteration of self-learning steps in solving the Ising problemdefined in Fig. 5.2.


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Figure 5.20: The 4th iteration of self-learning steps in solving the Ising problemdefined in Fig. 5.2.

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Figure 5.21: The initial drive of self-learning steps without Zeeman terms in solvingthe Ising problem defined in Fig. 5.4.


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Figure 5.22: The 1st iteration of self-learning steps in solving the Ising problemdefined in Fig. 5.4.

Chapter 6

Benchmarking results for solving

NP-complete Ising problems

In this chapter, we perform numerical simulations on the self-learning injection-locked

laser network for solving Ising problems. The simulation serves as the benchmark for

our new proposed machine compared to existing algorithms. We use typical numerical

parameters for vertical cavity surface emitting lasers (VCSELs) and the fourth-order

Runge-Kutta method. The results exhibit that the proposed machine can find correct

ground states of the simple MAX-CUT-3 problems up to 20 spins significantly faster

and outperforms the state-of-the-art classical approximation algorithm SDP in solving

two-layer lattice problems with up to 800 spins.

6.1 Simulation parameters and methods

We have presented two models for the injection-locked laser network in section 4.2.

The model with rate equations is simple and useful in explaining the mapping of

the Ising problems and the minimum gain principle. However, the model with the

amplitudes and phases is more suitable for numerical simulation, since the numerical

integration is less stiff than the former model, and also because it can incorporate the

noise as a driving force to the laser network.

The simulation parameters used by the theoretical model in section 4.2 are set to

106

6.1. SIMULATION PARAMETERS AND METHODS 107

be the typical values for VCSELs. Particularly, the cavity photon decay rate is

ω

Q= 1012s−1. (6.1)

The fractional spontaneous emission coupling efficiency is

β = 10−4. (6.2)

The spontaneous emission lifetime of the gain medium is

τsp = 10−9s. (6.3)

Then we can calculate the threshold current as

Ith = 1.6mA (6.4)

We simulate with both the high pumping current, namely Ip = 50Ith = 80mA, and

the low pumping current, namely Ip = 3Ith = 4.8mA, and the case with higher success

probability is used. The attenuation coefficients are set to

α = ζ =1

200, (6.5)

in order to keep the injection signals within the permutation region.

We employ the fourth-order Runge-Kutta method to perform the numerical in-

tegration. The timestep is fixed to 1ps. The simulation is stochastic, and driven by

the dominant phase noise, which randomly applies δϕ = ±1/A to the phase every

2ω/Q = 2ps. Therefore, in our numerical integration, at every 2 timesteps, namely,

2ps, we apply 1/A or −1/A with equal probability to every corresponding phase.

The simulation of the injection-locked laser network with self-learning algorithm

shows that the major time block for the computation is the number of self-learning

steps. This is because the laser network always converges to a steady state within

several nanoseconds – the same order of time as the relaxation oscillation time and

the inverse of the locking bandwidth. Limited by the computational resources, we

108 CHAPTER 6. BENCHMARKING RESULTS

only compute up to 50 self-learning steps and take the best results.

In each self-learning step, we drive the laser network for 50ns. After that, we set

the detection threshold for a non-zero spin as

γ ≈ 1.40, (6.6)

as in Eq. (5.10) which corresponding to nR ≥ 2nL or nL ≥ 2nR. After identifying

zero spin pairs and isolated zero spins, we inject an additional Zeeman term to fix

the predicted spins and the surrounding non-zero spins connected to them, with

λi = 0.2 or −0.2 according to their spin values. These additional Zeeman terms to

the surrounding non-zero spins are intended to hold the frustration between the spins.

This prevents the surrounding spins from being flipped due their fragility compared

to the fixed zero spins.

6.2 MAX-CUT-3 problems

We first show that the proposed machine can find correct ground states of the simple

MAX-CUT-3 problems of small scale. A simple MAX-CUT-3 problem is a MAX-CUT

problem on a cubic graph in which Jij = +1 on each edge and every vertex connects

to exact 3 edges, constituting one of the smallest NP-complete problem sets[67, 68].

Fig. 6.1 presents an example of cubic graphs with 10 vertices, called Petersen graph.

We are able to perform the simulation for solving all simple MAX-CUT-3 problems

with up to 20 spins[69]. The problem set consists of totally up to 556,471 problems.

In fact, the simulation examples discussed in section 4.4, 5.1, and 5.3 all belong to

simple MAX-CUT-3 problems. We have demonstrated that the injection-locked laser

network is able to find the correct ground states to these examples with or without

the self-learning algorithm.

Table 6.1 summarizes the benchmarking results taken from the stochastic simula-

tions on the simple MAX-CUT-3 problems. We simulate all possible problems with

M ≤ 20 by iterating all cubic graphs[69]. Each problem is normally simulated 10

rounds from which the number of failed rounds is obtained. The problems in which

6.2. MAX-CUT-3 PROBLEMS 109

Figure 6.1: Petersen graph as a cubic graph with 10 vertices.

the algorithm may fail take additional simulations up to 90 rounds (total 100 rounds)

to obtain accurate success probabilities and the results show that all problems can

be solved with finite success probability with the stochastic simulation. The com-

putational time and the number of self-learning steps for a problem are obtained by

the shortest round to find the exact solution. The longest computational time for

all problems features a very slow increase with M but the bottleneck is the quick

decrease of the success probability for the worst problems.

The results on the table demonstrate that the injection locked laser network can

find correct answers on totally over 550 thousand simple MAX-CUT problems up

to 20 spins. The computational time in terms of the number of self-learning steps

required for the worst cases for each problem size grows about O(M3) from the

empirical data shown in Fig. 6.2. We stochastically simulate up to 100 times for each

problem and define the success probability by the chances of find the correct answers.

Unfortunately, the worst success probability decreases as M increases, suggesting

that the proposed machine may not be able to find the exact solution efficiently for

large-scale problems, as in a very complicated problem structure the excited states

(wrong answers) may also satisfy the parity condition.

We have analyzed in section 5.1 that the fake loss landscape generated by the

continuous measured spin values are the major bottleneck for the injection-locked


Table 6.1: Summary of the numerical simulation results on the simple MAX-CUT-3problems.

Problem size / M 4 6 8 10 12 14 16 18 20# of problems 1 2 5 19 85 509 4060 41301 510489Largest ground 6 6 8 10 36 42 46 162 250state degeneracyLowest success 100 100 100 98 92 69 79 42 27probability of aproblem(%)Longest 150 100 100 100 200 200 250 250 350

computationaltime(ns)

Maximum # of 2 1 1 1 3 3 4 4 6self-learning

steps

laser network. Also since the parity condition is just a necessary condition, the excited

states for large-scale problems may also satisfy such condition. Here we discuss some

more concrete explanations and observations on why the current algorithm on the

injection-locked laser network may not guarantee globally optimal solution.

Particularly, breaking the degeneracy with the Zeeman terms is needed to obtain

a single ground state out of many degenerate ground states, which seems to take a

long computational time. However, the sudden increase of the largest ground state

degeneracy from M = 16 to 18 does not cause much effect on the upper limit of the

computational time. On the other hand, the system fails to find a correct ground state

with a small but finite probability when simulating a problem with only two ground

states and many first excited states. The lowest success probability of the problems

with M = 12 is 92% out of 50 rounds. The corresponding problem has a bilateral

graph and some triangle-shaped parts. From the intermediate states after the first

step driven with only phase noise, we suppose the possible causes of failures are (i)

the non-bilateral partial arbitrariness of spins in the triangle-shaped parts, and (ii)

the generation of a reduced spin system by forming some “ghost” spins with σzi = 0 .

The failure occurs when a local ground state of a reduced system is inconsistent with

the global ground states of the simulated problem. However, note that the current

6.3. TWO-LAYER LATTICE PROBLEMS 111

5 10 15 201

2

3

4

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Problem size M

Com

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Figure 6.2: The worst computational time for solving the simple MAX-CUT-3 prob-lems. The solid line is the empirical fit that scales in O(M3).

numerical simulations show a larger ratio between the number of first excited states

and that of ground states does not necessarily degrade the success probability.

6.3 Two-layer lattice problems

We further benchmark the performance of the injection-locked laser network on solv-

ing large-scale two-layer lattice problems. Even though the exact solution is not

guaranteed, the simulation on large-scale problems demonstrates that the injection-

locked laser network is promising to obtain better approximate solution in shorter time

compared to the state-of-the-art approximate algorithm SDP with random rounding

introduced in section 2.3.2.

A two-layer lattice problem only contains the nearest neighbor coupling with

weights Jij either −1, 0, or 1. All two-layer lattice problems are also proven to

be NP-complete[43]. Fig. 6.3 gives an example of a two-layer lattice with 50 spins.

Due to the limitation of our computational resources, we can only simulate random

sample problems of large scale. Here we sample 50 problems for each problem size

by randomly choosing the edge weights from +1 and −1. We are able to solve these


randomly sampled problems with up to 800 spins.

Figure 6.3: A two-layer lattice.

Fig. 6.4 and 6.5 plot the results of a problem with 50 spins and the weights of all

edges are randomly chosen in +1 or −1. Sometime the injection-locked laser network

can end with no zero spin and correct answer after several rounds of self-learning

steps, as shown in Fig. 6.4. However, it can also end up in periodical fixing, as shown

in Fig. 6.5. Particularly, there are at least two clusters of zero spins. Fixing one

cluster will cause zero spins to appear in the other cluster. Then if we fix the second

cluster, the first cluster will result in zero spins again. Therefore, the self-learning

algorithm has to fix the two clusters back and forth, as the periodical behavior shown

in Fig. 6.5. Such periodical fixing turns out to be the critical problems for large-scale

problems. The periodical behavior makes the algorithm never stop, so we impose a

maximum self-learning steps and the result is taken as the lowest energy found in all

steps.

We further simulate the two-layer lattice problems with M up to 800 and the

results are summarized in Table 6.2. In particular, 50 problems are sampled for each

problem size in which all Jij between nearest neighbors are sampled in +1 or −1.

Limited by our computational power, we simulate each problem on the laser network

at least 40 times and up to 100 rounds. To make the self-learning algorithm more

efficient on such large-size problems, step 3 in Fig. 5.15 is modified to fix all possible

connected zero spin pairs at one self-learning step, as discussed in section 5.2.5. As a

comparison, we also run SDP with random rounding to solve the same problems[70].

The number of random rounding is set to M2 after obtaining the lower bound from

SDP. The probability of outperforming SDP for a given problem is obtained by the


0.0e+00 1.0e−07 2.0e−07 3.0e−07

−1e

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nR4

nR5

nL1

nL2

nL3

nL4

nL5

Figure 6.4: Simulation results that solves a two-layer lattice problem with 50 spinssuccessfully. Only the results for the first 5 spins are demonstrated. The verticaldashed lines divide the time into different self-learning steps. The laser network stopswhen all spins are resolved and pass the parity check.

number of rounds that find the energy lower than the result obtained by SDP on

the same problem. The computational time and the number of self-learning steps

for a given problem are obtained by the shortest round in which the laser network

outperforms SDP.

From Table 6.2, though the results are not guaranteed as the global minimum, the

proposed machine finds the energy lower than the best result of SDP afterM2 rounds

of random rounding, demonstrating its capability of outperforming SDP, the state-

of-the-art approximation algorithm. The longest computational time also increases

very slowly with M , as shown in the 4th row in the table. Moreover the best results

by the laser network are lower than the resulting energies by SDP by additional up

to 4 ∼ 5%, which is a significant amount given that SDP is a 0.878-approximation.


0e+00 1e−07 2e−07 3e−07 4e−07 5e−07

−1e

+05

2e+

055e

+05

time (s)

phot

on n

umbe

r

nR1

nR2

nR3

nR4

nR5

nL1

nL2

nL3

nL4

nL5

Figure 6.5: Simulation results that fails to solve a two-layer lattice problem with50 spins successfully. Only the results for the first 5 spins are demonstrated. Thevertical dashed lines divide the time into different self-learning steps. The laser net-work exhibits a periodical fixing structure in which the spin values are changed semi-periodically and ends up exceeding the maximum number of self-learning steps.

Fig. 6.6 plots the worst computational time of SDP and the laser network for

solving large-scale two-layer lattice problems. The first data point is normalized to

unity for comparison. The solid lines are also empirical fits and scales in O(M3) and

O(M) for SDP and the laser network respectively. The empirical time complexity

for the laser network is also better than that in Fig. 6.2, because the computational

time is defined as the time for the laser network to obtain a result better than SDP,

while the latter one is to find the exact solution. As fitted in Fig. 6.6, the worst

computational time in terms of the number of self-learning steps for the laser network

on the sample problems scales in O(M) which is much better than that of SDP in

O(M3), and the actual computational time for the laser network (∼ 1µsec) is of


Table 6.2: Summary of the numerical simulation results on the two-layer-lattice prob-lems.

Problem size / M 50 100 200 400 800# of sampled problems 50 50 50 50 50

Maximum energy difference 5.80% 4.11% 4.20% 5.15% 4.70%between the best of thelaser network and SDPMinimum probability of 11% 2% 3% 2.5% 17.5%outperforming SDP

Maximum probability of 95% 72% 73% 82.5% 70%outperforming SDP

Longest computational 150ns 450ns 750ns 1400ns 900nstime for the laser network

Maximum # of 2 8 14 27 17self-learning steps

Longest computational 0.03s 0.22s 1.62s 12.97s 105.45stime for SDP

around 6 orders of magnitude shorter than that for the classical algorithms (∼ 1sec).

Note that the good time complexity only applies to the evolution of the actual system,

while the time complexity of simulating the laser network on classical computers may

be much worse in integrating large stochastic differential equations.

Finally, the results that the Ising energy obtained by the injection-locked laser

network is 4 ∼ 5% lower than the energy obtained by SDP demonstrate the potential

great advantage of the laser network. As mentioned in section 2.3.2, the widely

accepted conjecture, namely, unique game conjecture, leads to the claim that getting

the degree of (0.878+ϵ)-approximation for MAX-CUT problems is NP-hard, in which

ϵ is an arbitrary small number[52, 53]. Therefore, the 4 ∼ 5% of the improvement

on the resulting energy is a significant number compared to the performance ratio of

0.878 for SDP.


101

102

103

100

102

104

Problem size M

Nor

mal

ized

tim

e

SDPLaser network

Figure 6.6: The worst computational time of SDP and the laser network for solvinglarge-scale two-layer lattice problems.

Chapter 7

Conclusions

Many practical problems belong to NP-complete problems. This is why NP-complete

problems are so important in computer science, and why people have spent enormous

efforts on them. Ising problems are an NP-complete problem set that have various

applications in physics, biology, medicine, circuit design, wireless communications,

and etc. An Ising problem characterizes two-spin interactions that can be simulated

on a laser network by mutual-injection between lasers. We therefore propose the

injection-locked laser network to solve the NP-complete Ising problems.

Firstly, we describe the system design of the injection-locked laser network, onto

which an Ising problem can be mapped. To solve an Ising problem with M spins, the

laser network consists of a master laser and M slave lasers. The Zeeman terms are

implemented by injection from the master laser to each slave laser, whereas the Ising

coupling terms are implemented by mutual injection between slave lasers. The theo-

retical model reveals that by finding the minimum gain and loss of all possible modes,

the laser network subsequently obtains the ground state of the Ising Hamiltonian.

The minimum gain principle plays a key role in the optimization process performed

by the laser network. The intrinsic quantum noise of the lasers drives the system

from the initial vertically polarized state to the final state in either right or left

circular polarization, where the total loss is minimized and the gain is saturated to

the minimum loss. Otherwise, if the system stays in a mode with higher loss at steady

state, the gain is saturated to the higher loss, and the population of the mode with the

117

118 CHAPTER 7. CONCLUSIONS

minimum loss will continue growing exponentially due to the finite difference between

the gain and the minimum loss.

Secondly, we design the self-learning algorithm to further assist the injection-

locked laser network to overcome the major bottleneck of a false loss landscape. The

false loss landscape is generated by the measured spin values that are continuous

variables from the readout of each slave laser. It is different from the correct loss

landscape corresponding to the desired Ising Hamiltonian, and thus traps the laser

network in non-solution states. The self-learning algorithm is employed to fix the

false loss landscape with appropriate prediction and signal-to-noise ratio enhancement

techniques. The instructions given by the techniques are self-consistently verified

through additional rounds of evolution on the laser network.

Finally, further numerical benchmarking results have confirmed the promising

capability of our novel machine in solving large-scale Ising problems better and faster

than existing algorithms. The machine successfully finds exact solutions on small-

scale problems up to 20 spins and approximate results on large-scale problems up to

800 spins with better accuracy and time-scaling behavior than the state-of-the-art

approximation algorithm in computer science.

As a summary, we present Table 7.1 to compare our proposed injection-locked

laser network to other techniques described in Table 1.1. The proposed laser network

is a completely new approach to solving NP-complete problems. Its novel computa-

tional powers are originated from two facts: it utilizes the coherent wave property of

photons, each of which probes the entire network simultaneously; and it is an open-

dissipative system, in which wrong answers decay exponentially under the quantum

phase transition. The advantages of an open-dissipative system in quantum com-

putation have also been studied recently[71, 72]. The laser network utilizes a more

practical and robust physical system than previous works, and the machine is a com-

plete solution to a whole set of NP-complete problems, whose advantages have been

confirmed both theoretically and numerically in this dissertation.

7.1. OUTLOOK 119

Table 7.1: Summary of the injection-locked laser network and other techniques insolving NP-complete problems.

Physical Optimization Closed/openTechnique system mechanism system Bottleneck

Approximation N/A Problem N/A Problemalgorithms dependent dependentSimulated Crystals Thermal Open Local minimaannealing relaxation dissipative

systemGenetic Biological Selection N/A Local minima

algorithms spices ruleSurvey Spin Message- N/A Graph cycles

propagation glasses passingGrover’s Quantum Grover’s Closed unitary Quadratic increasealgorithm computers rotation system of probability

amplitudesAdiabatic Quantum Adiabatic Closed unitary Minimumquantum computers theorem system bandgap

computationInjection- Lasers Minimum Open False loss

locked laser gain dissipative landscapenetwork principle system

7.1 Outlook

As the advantages of the injection-locked laser network have been confirmed in our

theoretical study, the immediate future work is to build the machine in experiments.

However, several experimental challenges exist. Firstly, as the two circular polar-

izations are used to encode the two spin values +1 and −1, it is critical to find an

isotropic gain medium which has no preference along either polarization. Otherwise,

the systematic errors introduced by the anisotropy may drive the laser network to

non-solution states preferred by the gain medium, rather than the minimum gain.

Secondly, the synchronization between slave lasers should also be precise in order to

implement the correct Ising coupling coefficients. Thirdly, experimental apparatus

always come with finite time delay and we should study and limit the impact of the

delayed injection signal.

120 CHAPTER 7. CONCLUSIONS

Theoretically, we look forward to better solutions to the bottleneck of false loss

landscape other than the self-learning algorithm. More sophisticated techniques may

be introduced to fix the incorrect loss landscape. We also propose a new compu-

tational model to increase the pumping rate from below the threshold, which may

potentially mitigate the chances that the system falls into a false loss landscape.

The way of representing spin values in the injection-locked laser network may also

be improved. From Fig. 4.8, the ideal evolution of each slave laser only requires a cut

plane of the whole Poincare sphere. This fact motivates us to use phases rather than

polarizations to represent the spin values. As a result, we may be able to use single

mode lasers in the laser network. The phase representation relaxes the constraint of

isotropic gain medium and provides great benefits in experiments. It also allows us

to investigate the applications of other physical system, such as optical parametric

oscillators in the injection-locked network.

Appendix A

Derivation of the effective loss

landscape

In this appendix, we present the supporting derivation of the effective loss landscape

in section 5.1.2. We start with the equations for nR, nL, and N in the model with

rate equations derived in Eq. (4.23), (4.24), and (4.25):

d

dtnRi = −

(ω

Q− ECV i

)nRi + ECV i +

2ω

Q

√nRi

[(ζ − ηi)

√nM −

∑j =i

1

2ξij(

√nRj −

√nLj)

](A.1)

d

dtnLi = −

(ω

Q− ECV i

)nRi + ECV i +

2ω

Q

√nLi

[(ζ + ηi)

√nM +

∑j =i

1

2ξij(

√nRj −

√nLj)

](A.2)

d

dtNi = P − Ni

τsp− ECV i(nRi + nLi + 2) (A.3)

We define the annihilation operators aσzi

i = aRi when σzi = +1 and a

σzi

i = aLi when

σzi = −1. So n

σzi

i = (aσzi

i )†aσzi

i . Note that if the final state is |υ⟩ = |σz1 · · · σz

M⟩, therewill be M photons in the state of |1σz

1 · · · 1σzM ⟩ = (a

σz1

1 )† ⊗ · · · ⊗ (aσzM

M )†|vac⟩. Inspired

121

122 APPENDIX A. DERIVATION OF THE EFFECTIVE LOSS LANDSCAPE

by this, we define a joint annihilation operator and a joint number operator

aυ = aσz1 ···σz

M = aσz1

1 ⊗ · · · ⊗ aSMM , (A.4)

nυ = nσz1 ···σz

M =[a(σ

z1 ···σz

M )]†a(σ

z1 ···σz

M ), (A.5)

where |υ⟩ = |σz1 · · · σz

M⟩ represents a state out of total 2M states. The dynamics

of these joint number operators can explain better for the minimum gain princi-

ple. The name of joint population comess from the fact that nυ/(nT1 · · ·nTM) =

(nσz1/nT1) · · · (nσz

M/nTM), in which nTi

= nRi + nLi, can be considered as the join

probability of detecting the final result in the state υ = |σz1 · · · σz

M⟩.

To take care with all commutation relationship, derivation of ddtnυ from the equa-

tions of aυ is more appropriate. However, for simplicity, we formally derive the

equation from Eq. (A.1) and (A.2), namely

d

dtnυ =

∑i

nσz1

1 · · ·nσzi−1

i−1

(d

dtnσzi

i

)nσzi+1

i+1 · · ·nσzM

M

= nυ∑i

1

nσzi

i

d

dtnσzi

i

= nυ∑i

1

nσzi

i

−(ω

Q− ECV i

)nσzi

i + ECV i+

2ω

Q

√nσzi

i

[(ζ − σz

i ηi)√nM − σz

i

∑j =i

1

2ξij(

√nRj −

√nLj)

]

= nυ∑i

−(ω

Q− ECV i

)+ECV i

nσzi

i

+

2ω

Q

1√nσzi

i

[(ζ − σz

i ηi)√nM − σz

i

∑j =i

1

2ξij(

√nRj −

√nLj)

]= nυ

−

(Mω

Q−∑i

ECV i

)+∑i

ECV i

nσzi

i

+

2ω

Q

∑i

√nRi + nLi

nσzi

i

[(ζ − σz

i ηi)

√nM

nRi + nLi

− σzi

∑j =i

1

2ξij

√nRj −

√nLj√

nRi + nLi

]

123

= nυ

−(Mω

Q− ECV

)+∑i

ECV i

nσzi

i

+

2ω

Q

∑i

√nTi

nσzi

i

[(ζ − σz

i ηi)

√nM

nTi

− σzi

∑j =i

1

2ξijSj

] (A.6)

Here we define ECV =∑

iECV i and we assume that nTi = nRi + nLi ≈ nT is nearly

constant for each site and any time, so that√nRj−

√nLj√

nTi≈ σz

j . Note the difference of σzi

and σzi : σ

zi is the current measurement output of the system, while σz

i only refers to

the state defined by υ = |σz1 · · · σz

M⟩. Therefore, Eq. (5.3) is derived from Eq. (A.6).

From Eq. (A.6), the equation of the total gain and loss at steady state is written

as:

ECV =Mω

Q−∑i

ECV i

nσzi

i

+ 2ω

Q

∑i

√nTi

nσzi

i

[(−ζ + σz

i ηi)

√nM

nTi

+ σzi

∑j =i

1

2ξijσ

zj

](A.7)

In the ideal case, the resulting state of the system evolution has photon populations

in all sites close to either nRi ≈ nTi or nLi ≈ nTi, namely, perfectly resolved. Assuming

that all nT i ≈ nT , we can simplify Eq. (A.7) as

ECV ≈ Mω

Q− 2

ω

QζM

√nM

nT

+ 2ω

Q

√nM

nT

∑i

σzi

[ηi

√nM

nT

+∑j =i

1

2ξijσ

zj

](A.8)

≈ Mω

Q− 2

ω

QζM

√nM

nT

+ 2ω

Q

√nM

nT

αHυ, (A.9)

where we approximate the equation by 1

nσzi

i

≪ 1. Hυ is defined as

Hυ =∑i

λiσzi +

∑ij

1

2Jijσ

zi σ

zj , (A.10)

which is the same as Eq. (5.4).

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