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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.

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The Quest for Quantum Ants

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  1. The Quest for Quantum Ants QIP Seminar, July 2007 Yair Wiener

  2. Agenda • Classical Ants • Ants Turn Quantum • Search by Quantum Robots • Search by Quantum Random Walk

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

  4. 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

  5. Ant Inspired Algorithms * • Ant Colony Optimization (ACO) • Edge Ant Walk (EAW) • Vertex Ant Walk (VAW) (*) Partial list

  6. Typical Problems • Searching a graph (static targets) • Hunters (dynamic targets) • Combinatorial optimization (e.g traveling salesman problem) • Finding shortest path • And much more …

  7. In ACO pheromones attract ants 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)

  8. 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

  9. 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

  10. 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

  11. In VAW pheromones repel ants Vertex Ant Walk (VAW) * • v := u’s neighbor with minimal value of • 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

  12. 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)

  13. Comparison to random walk • Random Walk: O(n2) • VAW: O(n) n fully connected graph

  14. 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.

  15. 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

  16. Ants Turn Quantum

  17. 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)

  18. Schrödinger's Ant • We will pursue with the second direction. • Relevant work include quantum robots and quantum random walk.

  19. Quantum Robots

  20. “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 Robots Plus Environments”, Paul Benioff, Phys. Rev. A 58, 1998

  21. 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)

  22. 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

  23. Quantum Robot Model

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

  25. Grover algorithm (reminder) • Function f takes the value 0 on all elements except one, w • m iterations of Q corresponds to a rotation by mθin the 2 dimensional Hilbert space spanned by the orthogonal vectors |α> and |w>.

  26. 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 ( )

  27. Using Quantum Robot • 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.

  28. We need to preserve reversibility and unitarily of the dynamics Using Quantum Robot • 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

  29. 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

  30. 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 ?

  31. 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

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

  33. Divide-and-conquer algorithm

  34. 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:

  35. Divide-and-conquer algorithm • 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:

  36. Divide-and-conquer algorithm • 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

  37. 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

  38. Quantum Random Walk

  39. 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

  40. 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

  41. Discrete Quantum Random Walk • We will not measure the coin state between iterations • The interference causes radically different behavior than classical random walk

  42. Discrete Quantum Random Walk • The asymmetry (bias to the left) comes from the Hadamard coin • A symmetric coin * (*) “Quantum walks and their algorithmic applications”, A. Ambainis, Int. J. Quantum Inf.1, 507–518, 2003

  43. Discrete Quantum Random Walk

  44. Why do we need a coin state ? • 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 (*) “From quantum cellular automata to quantum lattice gases ”, D. Meyer, J. Stat. Phys. 85 (1996) 551-574

  45. 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

  46. The Model • Query transformation consists of two transformations • is applied to all for which and is applied to all for which • Z-local transformation * (*) “Quantum search of spatial regions”, S. Aaronson and A. Ambainis, In Proc. 44th Annual IEEE Symp. On Foundations of Computer Science (FOCS), pages 200-209, 2003

  47. The Model cont • The algorithm starts in a fixed starting stateand applies • The result is • Then we measure the final state

  48. Search by Quantum Random Walk * • Unperturbed “coin-flip” transformation • Perturbed “coin-flip” transformation • Final “coin-flip” transformation (*) “Coins Make Quantum Walks Faster”, A. Ambainis, J. Kempe and A. Rivosh, Proc. 16th ACM-SIAM SODA, p. 1099-1108 (2005)

  49. 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

  50. 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 Quantum Walk Search Algorithm

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