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. 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

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. 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 In VAW pheromones repel ants

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(n 2 ) VAW: O(n ) n fully connected graph

14. Comparison to random walk See example graph G 4 Complexity of reaching rightmost node: Random Walk: O(2 n ) VAW: O(n2 n ) (*) 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. 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 (*)

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

25. 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)

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. 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

28. 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

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. 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 ? (*) “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

33. Divide-and-conquer algorithm

34. 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

35. 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

36. 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

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. 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 ? (*) “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. 44 th Annual IEEE Symp. On Foundations of Computer Science (FOCS), pages 200-209, 2003

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

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

51. Grover as a Quantum Walk Grover search algorithm can be viewed as a random walk search algorithm on a complete graph Lets define

52. 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)

53. 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

54. 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

55. 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

56. Conclusion

57. Quantum Random Walk Grover (Quantum Robots) Classical Structured Graph (N x N grid) ??Unknown Graph