1. Quantum random walks and quantum algorithms Andris Ambainis University of Latvia

2. Part 1 Quantum walks as a mathematical object

3. Random walk on line Start at location 0. At each step, move left with probability ½, right with probability ½. -2012... Continuous time version: move left/right at certain rate.

4. Cont. time quantum walk Random walk: Quantum walk: Adjacency matrix:

5. Random walk on line State (x, d), x –location, d-direction. At each step, Let d=left with prob. ½, d=right w. prob. ½. (x, left) => (x-1, left); (x, right) => (x+1, right). -2012...

6. Quantum walk on line States |x, d , x –location, d-direction. -2012... “Coin flip”: Shift:

7. Classical vs. quantum Run for t steps, measure the final location. Distance:  (  t) Distance:  (t)

8. Semi-infinite walk Start at 0. At each step, move left with probability ½, right with probability ½. Stop, if we are at –1. Quantum version: project out the components at |-1, left  and |-1, right . 012...

9. Semi-infinite walk [A, Bach, et al., 01] What is the probability of stopping? Classically, 1. Quantumly, 2/ . With some probability, quantum walk “never reaches” –1. 012...

10. Finite walk [Bach, Coppersmith, et al., 2003] Start at 0. Stop at –1 or n+1. Classically, probability to stop at –1 is n/(n+1). Quantumly, it tends to 1/  2, for large n. 012... n Surprising, for two reasons

11. Probabilities to stop at -1 ClassicalQuantum Boundaries at –1 and n n/(n+1)  1/  2, for large n Semi-infinite1 2/  “Semi-infinite” is not limit of “large n” 1/  2 > 2/  Having a faraway border increases the chance of returning to -1

12. Explanation time location A second boundary reflects part of the state

13. Quantum walk on general graphs H – adjacency matrix of a graph.

14. Discrete quantum walk

15. Edges: |u, v . 1.“Coin flip”: 2.“Shift”:

16. Part 2 Applications of quantum walks

17. Quantum search on grids [Benioff, 2000]  N*  N grid. Each location stores a value. Find a location storing a certain value.

18. Grover’s search Find i for which x i =1. Questions: ask i, get x i. Classically, N questions. Quantum, O(  N) questions [Grover, 1996]. 0100... x1x1 x2x2 xNxN x3x3

19. Quantum search on grids [Benioff, 2000] Distance between opposite corners = 2  N. Grover’s algorithm takes steps. No quantum speedup.

20. Quantum search on grids [A, Kempe, Rivosh, 2004] O(  N log N) time quantum algorithm for 2D grid. O(  N) time algorithm for 3 and more dimensions.

21. Quantum walk on grid Basis states |x,y, , |x, y, , |x, y, , |x, y, . Coin flip on direction:

22. Quantum walk on grid Shift: |x, y,   |x-1, y,  |x, y,   |x+1, y,  |x, y,   |x, y-1,  |x, y,   |x, y+1, 

23. Search by quantum walk Perform a quantum walk with “coin flip”: C in unmarked locations; -I in marked locations. After steps, measure the state. Gives marked |x, y, d  with prob. 1/log N*. In 3 and more dimensions, O(  N) steps, constant probability. *Improved to const [Tulsi, 2008]

24. Element distinctness Numbers x 1, x 2,..., x N. Determine if two of them are equal. Well studied problem in classical CS. Classically: N steps. Quantumly, O(N 2/3 ) steps. 7921... x1x1 x2x2 xNxN x3x3

25. Element distinctness as search on a graph Vertices: S  {1,..., N} of size N 2/3 or N 2/3 +1. Edges: (S,T), T=S  {i}. Marked: S contains i, j,x i =x j. In one step, we can Check if vertex marked; or Move to adjacent vertex. {1,2} {1,3} {1,4} {1, 2, 3} {1, 2, 4} N 2/3 N 2/3 +1

26. Element distinctness as search on a graph Finding a marked vertex in M steps => element distinctness in M+N 2/3 steps. At the beginning, read all x i Can check if vertex marked with 0 queries. Can move to neighbour with 1 query. {1,2} {1,3} {1,4} {1, 2, 3} {1, 2, 4} A quantum walk finds a marked vertex in N 2/3 steps.

27. Hitting times Markov chain M, start in a uniformly random state. A marked state x. T – expected time to reach x. Theorem [Szegedy, 04] Given any symmetric Markov chain M, we can construct a quantum algorithm that finds a marked state in time O(  T)*. *May or may not apply to multiple marked states.

28. Testing matrix multiplication [Buhrman, Spalek 03] n*n matrices A, B, C. Does A*B=C? Classically: O(n 2 ). Quantum: O(n 5/3 ). Uses quantum walk on sets of columns/rows.

29. AND-OR tree ANDOR 1x11x1 2x22x2 3x33x3 4x44x4 5x55x5 6x66x6 7x77x7 8x88x8 ANDOR

30. Evaluating AND-OR trees Variables x i accessed by queries to a black box: Input i; Black box outputs x i. Quantum case: Evaluate T with the smallest number of queries. ANDOR 1x11x1 2x22x2 3x33x3 4x44x4

31. Results Full binary tree of depth d. N=2 d leaves. Deterministic:  (N). Randomized [SW,S]:  (N.753… ). Quantum? Easy q. lower bound:  (  N). ANDOR 1x11x1 2x22x2 3x33x3 4x44x4

32. [Farhi, Goldstone, Gutmann]: O(  N) time quantum algorithm in Hamiltonian query model

33. Flurry of improvements A. Childs, B. Reichardt, R. Spalek, S. Zhang. arXiv:quant-ph/0703015. A. Ambainis, arXiv:0704.3628. B. Reichardt, R. Spalek, arXiv:quant- ph/0710.2630.

34. Improvement I ANDOR ANDOR 1x11x1 2x22x2 3x33x3 4x44x4 5x55x5 6x66x6 Quantum algorithm for unbalanced trees

35. Improvement II O(  N) time Hamiltonian quantum algorithm O(N 1/2+o(1) ) query quantum algorithm [Farhi, Goldstone, Gutmann]: We can design discrete query algorithm directly.

36. [Childs et al.]: … Finite “tail” in one direction 0110

37. [Childs et al.]: … Basis states |v , v – vertices of augmented tree. Hamiltonian H, H- adjacency matrix of augmented tree.

38. [Childs et al.]: … 1 1 Starting state: Hamiltonian H, H – adjacency matrix

39. What happens? If T=0, the state stays almost unchanged. If T=1, the state scatters into the tree. … 0110 Surprising: the behaviour only depends on T, not x 1, …, x N.

40. More precisely… T=0: H has a 0-eigenstate with 0 amplitudes on x i =1 leaves. T=1: any 0-eigenstate of H has  (1/N) of itself on x i =1 leaves. … 0110

41. More precisely… T=0: H has a 0-eigenstate. T=1: All eigenvalues are at least 1/  N. … 0110 Time  1/min eigenvalue  O(  N)

42. From Hamiltonians to unitaries H 0 - AND-OR formula H 1 – extra edges for x i =1 H=H 0 +H 1 U=U 1 U 0

43. From Hamiltonians to unitaries … U 0 |  =-|  if H 0 |  = | ,  0. U 1 |v  =-|v  if v contains x i =1. 0-eigenstate of H  1-eigenstate of U 1 U 0

44. Handling unbalanced trees Weighted adjacency matrix H: H uv  0 if there is an edge between u,v. H uv depends on the number of vertices in subtrees rooted at u and v. [CRSZ]: apply Hamiltonian H. [A]: apply unitary U: U 0 |  =-|  if H|  = | ,  0.

45. Results (general trees) Theorem Any AND-OR formula of depth d can be evaluated with O(  Nd) queries. BCE91: Let F be a formula of size S, depth d. There is a formula F’, F=F’, 1.Size(F’)=O(S 1+  ), Depth(F’)=O(log S). 2.Size(F’)=, Depth(F’)= O(N 1/2+  ) quantum algorithm for any formula F

46. [Reichardt, Spalek] MAJ 1x11x1 2x22x2 3x33x3 4x44x4 5x55x5 6x66x6 7x77x7 8x88x8 9x99x9 MAJORITY tree: O(2 d ), optimal. Span programs

47. Summary: applications Quantum walks allow to solve: Element distinctness, Search on the grid, Matrix product verification. Boolean formula evaluation. Mostly via faster search for a marked location. Can we use quantum walks for fast sampling?

48. Search vs. formulas If no marked states, quantum walk stays in the start state. Otherwise, walk moves to marked states. If T=0, quantum walk almost stays in the start state. Otherwise, walk moves to a subtree that implies T=1. Marked states – local property T=1 – global property