1 / 48

Quantum random walks and quantum algorithms

Quantum random walks and quantum algorithms. Andris Ambainis University of Latvia. Part 1. Quantum walks as a mathematical object. Random walk on line. . . Start at location 0. At each step, move left with probability ½, right with probability ½. -2. -1. 0. 1. 2.

nate
Download Presentation

Quantum random walks and quantum algorithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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. -2 -1 0 1 2 Continuous time version: move left/right at certain rate.

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

  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). -2 -1 0 1 2

  6. Quantum walk on line ... ... -2 -1 0 1 2 • States |x, d, x –location, d-direction. “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. 0 1 2

  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. 0 1 2

  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. n 0 1 2 Surprising, for two reasons

  11. Probabilities to stop at -1 “Semi-infinite” is not limit of “large n” Having a faraway border increases the chance of returning to -1 1/2 > 2/

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

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

  14. Discrete quantum walk

  15. Edges: |u, v. • “Coin flip”: • “Shift”: Discrete quantum walk

  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 ... 0 1 0 0 • Find i for which xi=1. • Questions: ask i, get xi. • Classically, N questions. • Quantum, O(N) questions [Grover, 1996]. x1 x2 x3 xN

  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 logN) 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 ... 7 9 2 1 • Numbers x1,x2, ...,xN. • Determine if two of them are equal. • Well studied problem in classical CS. • Classically: N steps. • Quantumly, O(N2/3) steps. x1 x2 x3 xN

  25. Element distinctness as search on a graph • Vertices: S{1, ..., N} of size N2/3 or N2/3+1. • Edges: (S,T), T=S{i}. • Marked: S contains i, j,xi=xj. • In one step, we can • Check if vertex marked; or • Move to adjacent vertex. {1,2} {1, 2, 3} {1,3} {1, 2, 4} {1,4} N2/3 N2/3+1

  26. Element distinctness as search on a graph • Finding a marked vertex in M steps => element distinctness in M+N2/3 steps. • At the beginning, read all xi • Can check if vertex marked with 0 queries. • Can move to neighbour with 1 query. {1,2} {1, 2, 3} {1,3} {1, 2, 4} {1,4} A quantum walk finds a marked vertex in N2/3 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(n2). • Quantum: O(n5/3). • Uses quantum walk on sets of columns/rows.

  29. AND OR OR OR OR AND OR x1 x2 x3 x4 x5 x6 x7 x8 AND-OR tree

  30. AND OR OR x1 x2 x3 x4 Evaluating AND-OR trees • Variables xi accessed by queries to a black box: • Input i; • Black box outputs xi. • Quantum case: • Evaluate T with the smallest number of queries.

  31. AND OR OR x1 x2 x3 x4 Results • Full binary tree of depth d. • N=2d leaves. • Deterministic: (N). • Randomized [SW,S]: (N.753…). • Quantum? • Easy q. lower bound: (N).

  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. AND OR OR AND OR x1 x2 x3 x4 x5 x6 Improvement I Quantum algorithm for unbalanced trees

  35. Improvement II [Farhi, Goldstone, Gutmann]: O(N) time Hamiltonian quantum algorithm O(N1/2+o(1)) query quantum algorithm We can design discrete query algorithm directly.

  36. 0 1 1 0 … Finite “tail” in one direction [Childs et al.]:

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

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

  39. If T=0, the state stays almost unchanged. If T=1, the state scatters into the tree. 0 1 1 0 What happens? Surprising: the behaviour only depends on T, not x1, …, xN. …

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

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

  42. H1 – extra edges for xi=1 U=U1 U0 H0- AND-OR formula From Hamiltonians to unitaries H=H0+H1

  43. From Hamiltonians to unitaries U0|=-| if H0|=|, 0. U1|v=-|v if v contains xi=1. … 0-eigenstate of H  1-eigenstate of U1U0

  44. Handling unbalanced trees • Weighted adjacency matrix H: • Huv0 if there is an edge between u,v. • Huv depends on the number of vertices in subtrees rooted at u and v. • [CRSZ]: apply Hamiltonian H. • [A]: apply unitary U: U0|=-| 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’, • Size(F’)=O(S1+), Depth(F’)=O(log S). • Size(F’)= , Depth(F’)= O(N1/2+) quantum algorithm for any formula F

  46. MAJ MAJ MAJ MAJ x1 x2 x3 x4 x5 x6 x7 x8 x9 [Reichardt, Spalek] MAJORITY tree: O(2d), 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. 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 Search vs. formulas

More Related