quantum annealing and the schrödinger-langevin-kostin...

IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case

SLK Equation: Discrete CaseConclusions and Outlook

Quantum Annealing and the Schrödinger-Langevin-Kostinequation

Diego de Falco Dario Tamascelli

Dipartimento di Scienze dell’InformazioneUniversità degli Studi di Milano

IQIS Camerino, October 28th 2008

de Falco D., Tamascelli D. QA and SLK Equation



Outline

1 Introduction

2 The Schrödinger-Langevin-Kostin equation: Continuous caseThe SLK EquationToy Models

3 SLK Equation: Discrete CaseQuantum optimizationBloch oscillationsAnderson localization

4 Conclusions and Outlook




References

G. Jona-Lasinio, F. Martinelli, and E. Scoppola.

New approach to the semiclassical limit of quantum mechanics. i. multiple tunnelig in one dimension.Comm. Math. Phys., 80:223 – 254, 1981.

B. Apolloni, C. Carvalho, and D. de Falco.

Quantum stochastic optimization.Stoc. Proc. and Appl., 33:223–244, 1989.

G.E. Santoro and E. Tosatti.

Optimization using quantum mechanics: quantum annealing through adiabatic evolution.J. Phys. A: Math. Gen., 39:R393–R431, 2006.

M.D. Kostin.

On the Scrödinger-Langevin equation.J. Chem. Phys., 57(9):3589–3591, 1972.

R.P. Feynman.

Quantum mechanical computers.Found. Phys., 16(6):507–31, 1986.

D. de Falco and D. T.

Speed and entropy of an interacting continuous time quantum walk.J. Phys. A: Math. Gen., 39:5873–5895, 2006.

F. Dominiguez-Adame and V. Malyshev.

A simple approach to Anderson localization in one-dimensional disordered lattices.Am. J. Phys., 72:226–230, 2004.

H. Perets et al..

Realization of quantum walks with negligible decoherence in waveguide lattices.Phys. Rev. Lett., 100:170506, 2008.




Optimization problems and heuristic approach

Combinatorial optimization: find good approximations to solution(s) ofhard minimization problems.

Thermal simulated annealing1: temperature dependent random walk onthe space of all admissible solutions, endowed with a potential profiledependent on the cost function.

1Kirkpatrick et al., Science, 220, 671-680, 1983de Falco D., Tamascelli D. QA and SLK Equation



Quantum annealing

SA

QA

x

V(x)

Thermal jumps vs. Tunnelling




Quantum annealing

Use a Quantum Walk to explore the solution space: suggested by thebehaviour of the stochastic process qν(t) associated (G. Jona-Lasinio etal. 1981) to the ground state φν of the Hamiltonian

Hν = −ν2

2∂2

∂x2 + V (x) (1)

where V (x) encodes the cost function to be minimized.

100 200 300 400 500t

-4

-2

2

4

x

-4 -2 2 4x Markov chain with state space

determined by stableconfigurations.

Long sojourns around minima ofV (x)

Rare large fluctuations from oneminimum to another.




QA: Imaginary time evolution

Hovever:

The ground state of the Hamiltonian is seldom known.

Approximations are required.

Idea: Quantum Annealing (de Falco et al., 1989)1 Start from a suitable initial condition φtrial (x) . . .2 . . . let it evolve under the Hamiltonian semigroup e−tHν ,

Hν = −ν2

d2

dx2 + V (x)

for “some” time . . .3 . . . get an (unnormalized) approximation of φν .4 . . . decrease ν . . .5 . . . go back to step 2 until ν → 0 (slowly enough).




Adiabatic computation vs. Dissipative dynamics

Can we use viscous friction instead of an adiabatic change of theHamiltonian to reach the ground state?




The SLK EquationToy Models

Outline

1 Introduction








The SLK Equation

SLK (Kostin, 1972) derives from Heisenberg-Langevin equation (Ford,Kac, Mazur, 1965).

Represents the quantum analogue of classical motion with frictionalforce proportional to velocity (rough Drude-Lorentz model of Ohmicfriction).

If ψ(t , x) =pρ(t , x)etS(t,x) is a solution of the nonlinear, norm preserving,

SLK equation

i∂ψ(t , x)

∂t= −ν

2

2∂2ψ(t , x)

∂x2 + V (x)ψ(t , x) + βS(t , x) (2)

then it satisfiesddt〈 ψ(t) |Hν | ψ(t) 〉 ≤ 0, β ≥ 0.

In the following we will show how this dissipative evolution can drive asuitable initial condition ψ(0, x) to the ground state φν(t).





Outline

1 Introduction








Toy model 1: warm up and numerical check

Require

φν(x) = c+ exp„− (x − a)2

4σ2+

«+ c− exp

„− (x + a)2

4σ2−

«+ c0 exp

„− x2

4σ20

«to be

1 the ground state of an Hamiltonian Hν2 belonging to the eigenvalue 0.

Those requirements determine the potential

V (x) =ν2

2φν(x)

d2

dx2 φν(x).

Set the initial condition ψ(0, x) =exp

− (x+a)2

4σ2−

!(2πσ2

−)1/4





Toy model 1

VHxLt=0t=tmax

-4 -2 2 4

0 10 20 30 40 50t

0.02

0.04

0.06

0.08

0.10

0.12

0.14

EΨt

0 10 20 30 40 50t

0.8

0.85

0.9

0.95

1.

ÈXΨt ÈΦΝ\È2

〈 ψ(t) |Hν | ψ(t) 〉 decreaseswith time

the “vacuum overlap”|〈 ψ(t) | φν 〉|2 approaches thevalue 1

some probability mass“tunnels” from the leftmost(local) to the rightmost (global)minimum.

Since we know the ground state wecan use this class of examples tocheck properties of the dynamics(convergence, convergence speed)and of the numerical methods used.





Toy model 2

Double-well potential:

V (x) =

8><>:V0

(x2−a2+)2

a4+

+ δx , for x ≥ 0

V0(x2−a2

−)2

a4−

+ δx , for x < 0.(3)

(parametrization as in 2)

The local minimum of the potential is wider than the global one.

In this case the gound state is unknown and we use SLK dynamics tofind it.

2G.E. Santoro and E. Tosatti. Optimization using quantum mechanics: quantum annealingthrough adiabatic evolution. J. Phys. A: Math. Gen., 39:R393-R431, 2006.





Toy model 2

VHxLt=0t=tmax

-4 -2 2 4 6

0 10 20 30 40 50t

0.89

0.55

EΨt

-4 -2 2 4x

0.5

1.0

1.5

2.0

WtHxL

〈 ψ(t) |Hν | ψ(t) 〉 decreaseswith time

ψ(tmax , x) is a goodapproximation of the unknownground state: compare V (x)with the potentialW (t , x) = 1

2ψ(t,x)∂2

∂x2ψ(t , x) +

〈 ψ(t) |Hν | ψ(t) 〉 t → tmax , ofwhich ψ(tmax , x) is the groundstate belonging to theeigenvalue 0.




Quantum optimizationBloch oscillationsAnderson localization

Outline

1 Introduction








Quantum Optimization: General framework

To make the intuition developed so far available on the context ofquantum optimization (quantum annealing or adiabatic computation) wasthe main objective of the seminal paper on QA.3

V : Q → R: cost function;

underlying graph: G = (Q,E), E possible moves in the search: e.g.Q = Qn = {−1, 1}n: search on the hypercube with edges betweenvertices at 1 Hamming distance.

3B. Apolloni, C. Carvalho, and D. de Falco. Quantum stochastic optimization. Stoc. Proc. andAppl., 33:223-244, 1989.





A simple case

Here, we take the much simpler following example:

Q = Λs = {1, 2, . . . , s};E = {{i, j} : (i, j) ∈ Λs × Λs ∧ |i − j| = 1} .Search by an interacting continuous time quantum walk governed byh = − 1

2

Ps−1j=1 | j + 1 〉〈 j |+ | j 〉〈 j + 1 |+

Psj=1 V (j)| j 〉〈 j |.

Quantum search of this kind can suffer from two quantum effects:

Bloch oscillations

Anderson localization

but . . .





Viscosity as a resource

. . . both problems can benefit of the introduction of a certain amount ofviscous friction.





Outline

1 Introduction








Bloch oscillations

x

t

20 40 60 80 100

100

200

300

400

x

t

20 40 60 80 100

100

200

300

400

The effect on ballistic (in the sense of deFalco, D.T 2006) evolution of a linearpotential V (x) = −gx :

energy-momentum relation E(p) = 1− cos p determines Blochoscillations;this prevents the wave packet from approaching the point x = s wherethe minimum of V (x) is located;greedy (gradient) optimization hindered by the fact that, on a lattice,v(p) = sin p.





IDEA

Idea: add some friction to prevent first Brillouin zone crossing.





Discrete Kostin equation

Given the discrete equation

i∂ψ(t , x)

∂t= (h ψ)(t , x) + K (t , x)ψ(t , x) = (4)

= −12

(ψ(t , x + 1) + ψ(t , x − 1)) + V (x)ψ(t , x)| {z }h

+ K (t , x)ψ(t , x)| {z }dissipation

. (5)

the requirement

ddt〈 ψ(t) |h| ψ(t) 〉 = −

s−1Xx=1

(K (t , x + 1)− K (t , x)) sin (S(t , x + 1)− S(t , x)) ,

(6)

is saftisfied, for example, if

K (t , x) = β

xXy=2

sin (S(t , y)− S(t , y − 1)) (7)





Bloch oscillations

Linear potential:

x

t

20 40 60 80 100

100

200

300

400

g = 0, β = 0;s = 100, ε = 17g0 = 2/s,0 ≤ t ≤ 4s.

Linear Potential + Viscous force:

x

t

20 40 60 80 100

100

200

300

400

x

t

20 40 60 80 100

100

200

300

400

100 200 300 400t

0.2

0.4

0.6

0.8

1.0

100 200 300 400t

0.2

0.4

0.6

0.8

1.0

g = 3g0, β = 2g0 g = 3g0, β = 4g0:





Outline

1 Introduction








Anderson localization

x

t

20 40 60 80 100

500

1000

1500

2000

x

t

20 40 60 80 100

100

200

300

400

x

t

20 40 60 80 100

500

1000

1500

2000

500 1000 1500 2000t

0.2

0.4

0.6

0.8

1.0

Random Gaussiannoise:2σ0 = 2 (10/s)3/2.

Linear potential:g = 3g0.

Kostin friction: β = 4g0.

pseudo-ballistic motionis much more stablethan the truly inertialone with respect to theonset of Andersonlocalization.




Conclusions

Bloch oscillations and Anderson localization can hinder a completesearch of the solution space of the optimization problem.

“Viscous” friction of Kostin suppresses both these effects.

Bloch oscillations are suppressed by preventing the wavepacketmomentum from crossing the first Brillouin zone.

Anderson localization is avoided by probability percolation through theirregular potential profile.




Outlook

The same framework developed in this note for the combinatorialoptimization metaphor can be used, with minor changes, to describe anexcitation travelling along a spin chain or a light pulse propagatingthrough a waveguide lattice.4

SLK dynamics can be exploited also in these fields.Increase the fidelity of state transmission, in presence of imperfectionsalong a spin chain, by applying a “tension” at both ends of it.5

The sole convergence toward the ground state could, instead, findapplications in all-optical switching of light in waveguide arrays 6: theinjected light pulse can be steered toward a given position by a suitabletuning of the thermal gradient which determines the potential profile ofthe lattice.So far we considered the dissipative part. Inclusion of fluctuations isrequired.Extension to general graphs. First step: Hypercube.

4H. Perets et al.. Phys. Rev. Lett., 100:170506, 2008.5R.P. Feynman. Quantum mechanical computers. Found.Phys., 16(6):507-31, 1986.6D.N. Christodoulides, F. Lederer, and Y. Silberberg. Discretizing light behaviour in linear and

nonlinear waveguide lattices. Nature, 424:817-823, 2003de Falco D., Tamascelli D. QA and SLK Equation



Thank You!


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