The Quest for Quantum Ants

QIP Seminar, July 2007

Yair Wiener

Agenda

Classical Ants Ants Turn Quantum Search by Quantum Robots Search by Quantum Random Walk

Classical Ants

“Go to the ant, thou sluggard; consider her ways, and be wise”

Biological Ants

Ants (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails

If other ants find such a path, they are likely to follow the trail, returning and reinforcing it if they eventually find food

Ant Inspired Algorithms *

Ant Colony Optimization (ACO) Edge Ant Walk (EAW) Vertex Ant Walk (VAW)

(*) Partial list

Typical Problems

Searching a graph (static targets) Hunters (dynamic targets) Combinatorial optimization (e.g traveling

salesman problem) Finding shortest path And much more …

Ant Colony Optimization (ACO) *

a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs

(*) Introduced by Marco Dorigo in his PhD thesis (1992)

In ACO pheromones attract ants

ACO framework (quick overview)

The pheromone model induce probability distribution over solution space

Multiple solutions (ants) are sampled and, optionally, locally optimized

Pheromone model is updated according to solutions (“good” solutions increase local probability)

Repeat sampling solution space and update pheromone model

Searching a Graph

Consider memoryless agent that searches a graph G(V,E) for food

Each agent (ant) has the ability to leave pheromone traces on vertices and sense the smell

Pheromone traces dissipate over time

Formalism

A vertex v at time t is marked by the pair :

where is the number of marks left on v, and is the time of the most recent mark left

)(vt)(vt

))(),(( vv tt

Vertex Ant Walk (VAW) *

1. v := u’s neighbor with minimal value of

2.

3.

4.

5. go to v

(*) “Efficiently Searching a Graph by a Smell-Oriented Vertex Process”, A. Wagner et al, Annals of Mathematics and Artificial Intelligence 24 (1998) pp. 211-223

(.))(.),( 1)(:)( uu

tu :)(1: tt

In VAW pheromones repel ants

Some VAW Results

Theorem 1: Denote by d the diameter of G, and by n the number of vertices. Then after at most nd steps the graph G is covered.

Searching graph G is O(nd)

Comparison to random walk

Random Walk: O(n2) VAW: O(n )

n fully connected graph

Comparison to random walk

See example graph G4

Complexity of reaching rightmost node:

Random Walk: O(2n) VAW: O(n2n )

(*) The graph is from “An example of the difference between quantum and classical random walks”, A.Childs et al, Quantum Information Processing, 1:35, 2002.

Summary

Ant inspired algorithms can find approximations to NP-hard problems

VAW can search any unknown graph in complexity O(nd)

The introduction of pheromones can improve search performance over random walk

Ants Turn Quantum

How Can We Turn Ants Quantum ?

Leave pheromones in quantum states (exploit quantum communication between ants)

Put each ant in superposition (travel different directions at the same time)

Schrödinger's Ant

We will pursue with the second direction. Relevant work include quantum robots and

quantum random walk.

Quantum Robots

Quantum Robot

(*) “Quantum Robots Plus Environments”, Paul Benioff, Phys. Rev. A 58, 1998

“A quantum robot is a mobile quantum system, including an on board quantum computer and needed ancillary systems, that interact with an environment of quantum systems”

Paul Benioff, 1998 (*)

Quantum Robot

Quantum Robot model is a generalization of previous work on quantum computers with interactions with the environment (noise effects, data base searching and quantum oracle computing)

Quantum Robot Model

Quantum Robot consist of– On board quantum computer– Memory system (m)– Output system (o)– Control qubit (c)

Task Dynamics– Alternating computation and action phases

Quantum Robot Model

aod

oda

cqrx

qrxc

n

cacc

cqrom

PP

PP

n

PP

iixdjn

)0()(

)(

1,0,||||)(

10

Space Search with Quantum Robot *

0Y

(*) “Space Searches with a Quantum Robot”, Paul Benioff, AMS Contemporary Math Series, Vol 305, 2002

N

N

0X

Function f takes the value 0 on all elements except one, w

m iterations of Q corresponds to a rotation by mθ in the 2 dimensional Hilbert space spanned by the orthogonal vectors |α> and |w>.

Grover algorithm (reminder)

NQ

xN

PPIIQ

xN

wx

ww

x

21cos,

cossin

sincos

|1

1|

)21)(21(

|1

Can we use Grover algorithm ?

Can we directly use Grover algorithms to solve the grid search problem in O(N) ?

The problem is that for efficient implementation of the algorithm it is required to determine, in small number of steps if x = w

In the grid search problem we don’t have access to the phase oracle ( )wI

Initial state: Copy m state onto L: Computation Phase:

– If X > 0: – Else if Y > 0:– Else test presence of s at the robot location and “record” it by

changing memory state phase.– Go back to the origin following the same path and change

output state to |dn> upon arrival.

Using Quantum Robot

QRcoLm dnYX 0,0|1||0|,|

QRcoLm dnYXYX 0,0|1||,|,|

ccooLL xYXYX 0|1|,|?|,,1|,|

ccooLL yYY 0|1|,|?|,1,0|,0|

Action phase– The quantum robot moves one lattice site according to the

output state direction– Upon arrival back to origin transfers motion to some ballast

system

Using Quantum Robot

We need to preserve reversibility and unitarily of the dynamics

Using Quantum Robot

We start with a quantum robot with the initial state:

After the quantum robot returns to the origin the state is:

The complexity of getting is

1

0,

,|1 N

YXmm YX

N

m

YXYXmf YXYX

N 00,

,|,|1

00

f )log( NNO

Using Quantum Robot

Using quantum robot for evaluation of the phase oracle in Grover search algorithm results in overall complexity of

The advantage of quantum, over classical searching is lost for 2 dimensional regions

What about d dimensional regions ?

)log( 2 NNO

)log(1

2 NNOd

Have we missed anything ?

Our discussion ignored the entanglement problem Entanglement occurs because the unitary dynamics

is reversible and the number of steps needed to complete the search task is different for different component states of

Grover algorithm requires the removal of this entanglement

Benioff claimed that it is improbable that Grover algorithm will be used to speed up spatial 2D search

m

Scott Aaronson and Andris Ambainis have shown in 2003 that Benioff’s claim is mistaken *

Searching 2-dimensional graph can be done in

And searching d-dimensional graph (d > 3) can be done in

Is it the end of the road ?

)log( 2

3

NNO

(*) “Quantum search of spatial regions”, S. Aaronson and A. Ambainis, In Proc. 44 th Annual IEEE Symp. On Foundations of Computer Science (FOCS), pages 200-209, 2003

)( 2

d

NO

Divide-and-conquer algorithm

Partition the region into squares Travel from start vertex to any setsquare C: Search C classically and return to start vertex: Applying Grover algorithm on C’s results: Overall search complexity:

Divide-and-conquer algorithm

)(NO

)(NO

)( NO

)( 2

3

NO

N

Now we can partition the region into squares Travel from start vertex to any setsquare C: Search C using previous technique: Applying Grover algorithm on C’s results: Overall search complexity:

Applying this technique recursively we get:

Divide-and-conquer algorithm

3

2

N

)()( 2

3

3

2

NONO

)(NO

)()( 3

1

2

1

3

2

NONO

)( 3

4

NO

)(NO

The problem is that, with each additional layer of recursion, the robot needs to repeat the search more often to upper bound the error probability

Amplitude amplification approach is used to overcome this issue and achieve the improved bounds

Divide-and-conquer algorithm

Summary

The introduction of physical constrains to quantum computations yields interesting results

Quantum robot: dynamic quantum system with alternating computation and action phases

Grover algorithm can indeed speed up spatial search

2D grid can be searched in using Grover algorithm and quantum robots

)log( 2

3

NNO

Quantum Random Walk

Discrete Quantum Random Walk

We will start with one dimensional quantum walk Let be the Hilbert space spanned by the position

of the particle

Let be the ‘coin’-space spanned by two basis states

States of the total system are in the space

}1...0:{| NiiH p

pH

cH

}|,{| cH

pc HHH

Discrete Quantum Random Walk

The conditional translation of the system can be described by the following operator

The unitary transformation C is very arbitrary An example of coin is Hadamard coin H Measuring the coin state after each iteration of

removes the correlation between positions and we obtain the classical random walk

ii

iiiiS |1||||1|||

)( ICS

Discrete Quantum Random Walk

We will not measure the coin state between iterations

The interference causes radically different behavior than classical random walk

3||1||2|1||3||22

1

2||0|||2||2

1

1||1||2

1|

)(

U

U

U

start

IHSU

Discrete Quantum Random Walk

The asymmetry (bias to the left) comes from the Hadamard coin

A symmetric coin *

||2

1|

||2

1|

H

H

1

1

2

1

i

iW

(*) “Quantum walks and their algorithmic applications”, A. Ambainis, Int. J. Quantum Inf. 1, 507–518, 2003

Discrete Quantum Random Walk

)( IHSU )( IWSU

Lets look on quantum random walk on a single line

David Meyer have shown (*) that the transformation U defined by the above equation is unitary only if

Why do we need a coin state ?

1||1|| ncnbnan

1||1||1|| cba

(*) “From quantum cellular automata to quantum lattice gases ”, D. Meyer, J. Stat. Phys. 85 (1996) 551-574

The Model

Given undirected graph Each vertex v stores a variable At one step an algorithm can examine the current

vertex or move to a neighboring vertex The algorithm is a sequence of unitary

transformations on a Hilbert space

),( EVG }1,0{va

vi HH

The Model

Query transformation consists of two transformations

is applied to all for which and is applied to all for which

Z-local transformation *

iU),( 10

ii UU

IU i 0 vH i | 0va

IU i 1 vH i | 1va

)()|(| vii HHvU

(*) “Quantum search of spatial regions”, S. Aaronson and A. Ambainis, In Proc. 44 th Annual IEEE Symp. On Foundations of Computer Science (FOCS), pages 200-209, 2003

The algorithm starts in a fixed starting stateand applies

The result is Then we measure the final state

The Model cont

start|

tUU ....,,1

startttfinal UUU |....| 11

Search by Quantum Random Walk *

Unperturbed “coin-flip” transformation

Perturbed “coin-flip” transformation

Final “coin-flip” transformation

N

d

id

ICC

id

sIssC

0

10 |

1|,||2

dIC 1

||)(|||)|( 1010 vvCCCvvCvvICC

(*) “Coins Make Quantum Walks Faster”, A. Ambainis, J. Kempe and A. Rivosh, Proc. 16th ACM-SIAM SODA, p. 1099-1108 (2005)

Search by Quantum Random Walk

S is a shift controlled by the coin register

Where is a permutation of the d basis states of the coin space

The “marked walk” operator

xixiS ~|)(||:|

CSU

Quantum Walk Search Algorithm

d

i

N

x

xidN 1 1

0 ||1

|

0|

Quantum Walk Search Algorithm

• Initialize the quantum system in the uniform superposition

• Do T times: Apply the marked walk• Measure the position register• Check if the measured vertex is the marked item

U

Grover as a Quantum Walk

Grover search algorithm can be viewed as a random walk search algorithm on a complete graph

Lets define

|)|2(

|||:|

001 vvICCCC

ijjiS

vs

ss

vv

IIC

ssIPII

vvIPII

|)|2(2

|)|2(2

Grover as a Quantum Walk

Now the random walk based algorithm is

The random walk gives exactly Grovers algorithm on both coin space and the vertex space (at the expense of factor 2 in the number of applications)

sIIsIIsIIsIISU

sIsIsIsISCSU

ss

vssvsvvs

svvs

|||||

||||||

|||

02

00

0

Searching a 2D grid

The choice of the coin transformation (or permutation ) is crucial for the performance of the random walk

Lets define two shift operators

1,||,||1,||,||

1,||,||1,||,||

,1||,||,1||,||

,1||,||:,1||,||:

yxyxyxyx

yxyxyxyx

yxyxyxyx

yxyxSyxyxS mff

The quantum walk can search N x N grid in steps

The quantum walk can search N x N grid in at least steps

Searching a 2D grid

CSU ff

CSU m )log( NNO

)( 2N

Summary

Quantum random walk exhibit substantially different behavior than classical random walk

The performance of quantum random walk as search algorithm highly depends on the coin transformation

2D grid can be searched in using quantum random walk

)log( NNO

Conclusion

Conclusion

ClassicalGrover (Quantum Robots)

Quantum Random Walk

Structured Graph

(N x N grid)

Unknown Graph

??

)log( NNO)( 2NO )log( 2

3

NNO

)(ndO