The Quest for Quantum Ants
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Transcript of The Quest for Quantum Ants
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
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Space Search with Quantum Robot *
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(*) “Space Searches with a Quantum Robot”, Paul Benioff, AMS Contemporary Math Series, Vol 305, 2002
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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)
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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
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QRcoLm dnYXYX 0,0|1||,|,|
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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
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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 ?
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)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 ?
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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
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d
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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
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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
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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
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cH
}|,{| cH
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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
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)( ICS
Discrete Quantum Random Walk
We will not measure the coin state between iterations
The interference causes radically different behavior than classical random walk
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Discrete Quantum Random Walk
The asymmetry (bias to the left) comes from the Hadamard coin
A symmetric coin *
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1|
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(*) “Quantum walks and their algorithmic applications”, A. Ambainis, Int. J. Quantum Inf. 1, 507–518, 2003
Discrete Quantum Random Walk
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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 ?
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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
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(*) “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
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i
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x
xidN 1 1
0 ||1
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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
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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)
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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
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1,||,||1,||,||
,1||,||,1||,||
,1||,||:,1||,||:
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yxyxyxyx
yxyxyxyx
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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
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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