quantum algorithms for moving-target tsp

8/3/2019 Quantum Algorithms for Moving-Target TSP

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Quantum Algorithms for

Moving-Target TSP

Prof. Rushen Chahal

Prof. Rushen Chahal


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Two Slit Experiment

Bullets

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Two Slit Experiment

Sound Waves

Prof. Rushen Chahal


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Electrons

Two Slit Experiment

Prof. Rushen Chahal


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Two Slit Experiment

Observing Electrons

Prof. Rushen Chahal


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Basic Quantum Computation Qubit - can be 1, 0 or both 1 and 0

|x> - number in Quantum Computer

Superposition of states:

Where:

!

12

0

N

i

iisa 1

12

0

2!

!

N

i

ia

Prof. Rushen Chahal


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Examples

12

10

2

1

112110

2101

2100

21

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Representation n qubits: 2nx1 matrix represents the state:

|0> would be represented by

|1> would be represented by

Equal superposition would be

-

1

0

-

0

1

-

2

12

1

Prof. Rushen Chahal


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Changing States Unitary transformations change states

Unitary matrix:

conjugate transpose = inverse

1! AA

T

Prof. Rushen Chahal


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ExampleHadamard Transform

-

2

1

2

121

21

-0

1

-

2

12

1

!

!

-

10

01

-

2

1

2

12

1

2

1

-

2

1

2

12

1

2

1

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Example: CNot Gate

-

01

10

-

0100

1000

0010

0001

Not Gate:

-

b

a

-

a

b!

-

d

c

b

a

!

-

c

d

b

a

CNot Gate:

11

10

01

00

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Visual Representation

00 102

100

2

1

112

1002

1 012

1102

1

p

p

p

H

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Grovers Search Algorithm Search a phone book for a specific number

without rearranging the numbers.

Idea: magnify amplitude of the choosen number:

Flip the amplitude of the selected item and

rotate all amplitudes around the average

Repeat this until the selected items probabilityof being read is greater than 1/2

Prof. Rushen Chahal


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Graphical Representation

Original Amplitudes Negate Amplitude

Average of all Amplitudes Flip all Amplitudes around Avg

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Time Complexity Normal search requires N/2 steps

Grovers Algorithm takes steps NO

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Traveling Salesperson Problem

a

b

cd

e

10

38

6

1211

4

25

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Origin

Moving-Target TSP

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3

2

1

4

Origin

Moving-Target TSP

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Moving Target TSP is Intractable

a

b

cd

e

10

38

6

1211

4

25

(V=0)

(V=0)(V=0)

(V=0)

(V=0)

NP-Hard

Classical TSP is NP-Complete

Classical TSP reduces to Moving-Target TSP

Prof. Rushen Chahal


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NP-Complete Contained in NP:

1. Decision version: path with time < T?

Non-deterministically travel all paths. If one exists

with time < T, return TRUE. Else, return FALSE

2. Optimization version: what is min-time path?

Upper-bound T with initial random path. Then,binary search the range by testing T/2, T/4, etc. to

find optimal the minimum- time path

Prof. Rushen Chahal


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Quantum Computing Solution Step 1 - Traverse every possible path

Step 2 - Search through paths superposition

to find a shortest path

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Superposition for

Hamiltonian Paths T. Rudolph

superposition of cubic bipartite graph in linear time

Cubic - all nodes have degree 3

Bipartite

nodes are partitioned into two groups

each node is only adjacent to nodes in other group

Prof. Rushen Chahal


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Example

00000000000010001000

34311001243111113131000021310110

34211111242110013121011021210000

4311011131001042111011210100

311010211100

1

4

2

3

11000

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Potential Problems Problem 1:

Extract the Hamiltonian paths

Problem 2:

Find cycles, not paths

Problem 3:

Only works for cubic graphs

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Solution to Problem 1 How to extract only paths containing all 1s

in first register?

Solution: Use Grovers Algorithm!

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Solution to Problem 2 Use black box for Hamiltonian paths to

solve for Hamiltonian cycles?

Solution:

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Solution to Problem 3 Problem 3: non-cubic graphs

Solution:

Make all nodes have the same degree

Degree must be a power of 2

Algorithm when all nodes have degree 2i

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Step 1Give all nodes same degree of 2x

Graph G has n nodes

Find node with largest degree D

Find x where xxD 22 1 ee

n+x-n(mod x)

Groups of x nodes

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Step 1 (continued)

Go through the graph G node by node, and

go through the new nodes set by set:

G

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1101

4

11000

4

10100

4

10010

4

1

H

H

11004

11000

4

10100

4

10010

4

1

H

11004

11000

4

10100

4

10000

4

1

H

10002

10000

2

10000

Algorithm Quantum transition for nodes with 2x degree:

Control bit

11014

11000

4

10100

4

10010

4

1

10014

11000

4

10100

4

10010

4

1

00014

11000

4

10100

4

10010

4

1

Prof. Rushen Chahal


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Observations

Algorithm works for all graphs

May double # of nodes

Takes O(n) steps

Search first register for all 1s

Added nodes do not affect solution

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Extension to TSP

Need another register:

Large enough to hold longest path

At each step, add edge weight into register

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Iteration

All paths of length n & their weights

Run algorithm for Hamiltonian cycles Look for all 1s in the first register

Superposition of all Hamiltonian cycles

Grovers algorithm on sum register finds min

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Extension to Moving-Target TSP Moving Target TSP

Each node has a velocity

Find the minimum-weight round trip

v2x

v2yv1x

v1y

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Moving-Target TSP Solution

Add a time register Track the total time elapsed so far

Each step: calculate time to reach next node

tvv

Unknowns

MaxVvv

atvntv

atvntv

yx

yx

yyyy

xxxx

:

1

1

222!

!

!

v1x

v1y

MaxV

yxaa ,

yx

nn ,

Prof. Rushen Chahal


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Determining the Optimal Path Add time to reach next node to total sum

Find the minimum, similar to TSP

Added time complexity is linear

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Summary

Extended Hamiltonian path algorithm for

cubic bipartite graphs [Rudolphs]

For TSP and Moving-Target TSP, paths

superposition can be obtained in linear time

Grovers search algorithm works in time

SQRT of # of objects in superposition 2n different paths: total time is O(2n/2 )

Prof. Rushen Chahal


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Future Work Faster algorithm for (Moving-Target) TSP

Improve Grovers search algorithm

P=NP for quantum computation?

Prof. Rushen Chahal


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Circuit

a,b,c are the nodes adjacent to I A is the register keeping track of which nodes

have been traversed

V

i

ab

c

ab

c

j

A

j+1

Prof. Rushen Chahal


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Finding the Minimum Measure the lengths register (M)

Create superposition again

Now search for all paths < M

On average, do this log(k) times

k is the number of items in the superposition

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Sound WavesBullets

Electrons Observed Electrons

Prof. Rushen Chahal

quantum algorithms for moving-target tsp

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