Quantum algorithms with polynomial speedups

1. Quantum algorithms with polynomial speedups Andris Ambainis University of Latvia

2. Search [Grover, 1996] ? ? ? ? ? ... • N objects; • Find an object with a certain property. Conventional computer: N objects Quantum computer: √N objects Useful for many computational tasks

3. Does this graph have a cycle that contains every vertex? Hamiltonian cycles • Hamiltonian cycles are: • Easy to verify; • Hard to find (too many possibilities). NP-complete problem.

4. Quantum algorithm • N objects = N possible cycles. • Classical search: N steps. • Quantum search: √N steps. ? =

5. Search by a quantum walk

6. Search by random walk • Search space with a structure. • Task: find a state with some property. • Walk randomly, according to a rule that uses the structure of search space. 1 3 2 4 5 6

7. A quantum particle moving around this state space. Quantum walk with two transition rules: “usual” for unmarked vertices; “special” for marked. Quantum walk 1 3 2 4 5 6 Particle “drifts” toward the marked states.

8. Random vs. quantum walks • [Szegedy, 2004] If a classical random walk* finds a marked state in T steps, a quantum walk finds it in O(√T) steps. • Generalizes Grover’s search by using the structure of the search space. *that satisfies some constraints.

9. Element distinctness (A, 2004) 28 12 18 76 96 82 94 99 21 78 88 93 39 44 64 32 99 70 18 94 82 92 64 95 46 53 16 35 42 72 31 40 75 71 93 32 47 11 70 37 78 79 36 63 40 69 92 71 28 85 41 80 10 52 63 88 57 43 84 67 57 31 98 39 65 74 24 90 26 83 60 91 27 96 35 20 26 52 95 65 66 97 54 30 62 79 33 84 50 38 49 20 47 24 54 48 98 23 41 16 66 75 38 13 58 56 86 34 73 61 73 21 44 62 34 14 51 74 76 83 37 90 58 13 10 25 29 25 56 68 12 11 51 23 77 68 72 43 69 46 87 97 45 59 14 30 19 81 81 49 60 85 80 50 61 59 89 67 89 29 86 48 22 15 17 55 36 27 42 55 77 19 45 15 53 22 91 87 17 33 Task: find two equal numbers.

10. 31 40 75 71 93 32 47 11 70 37 78 79 36 63 4048 98 23 41 16 66 75 3827 42 55 77 19 45 15 53 22 9137 90 58 13 10 25 29 25 56 68 12 11 51 23 7715 17 Classical: N steps. Quantum: ~N2/3 steps. Element distinctness

11. Triangle finding [Magniez, Santha, Szegedy, 03] • Graph G with n vertices. • n2 variables xij; xij=1 if there is an edge (i, j). • Does G contain a triangle? • Classically: O(n2). • Quantum: O(n1.3).

12. Matrix multiplication [Buhrman, Špalek, 05] • A, B, C – n*n matrices. • Given A, B and C, we can test AB=C in: • O(n2) steps by a probabilistic algorithm; • O(n5/3) steps by a quantum algorithm.

13. Quantum simulated annealing[Somma, Boixo, Barnum, 2008] • Simulated annealing is a general heuristic for solving optimization problems. • [Somma, Boixo, Barnum, 2008]: quantum version of simulated annealing, with a quadratic speedup.

14. Quantum speedups are very common

15. Evaluating Boolean formulas

16. AND OR OR x1 x2 x3 x4 [Farhi et al., 07] • AND-OR formula of size M. • Variables accessed by queries: ask i, get xi.

17. Vertices = chess positions; Leaves = final positions; xi=1 if the 1st player wins; At internal vertices, AND/OR evaluates whether the player who makes the move can win. OR x1 x2 Motivation How well can we play chess if we only know the position tree?

18. 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? FGG: O(√N) quantum algorithm

19. 0 1 1 0 … … … … Infinite line in two directions [Farhi, Goldstone, Gutmann]:

20. [Farhi, Goldstone, Gutmann]: • Basis states |v, v – vertices of augmented tree. • Hamiltonian H, H-adjacency matrix of augmented tree. … …

21. [Farhi, Goldstone, Gutmann]: • Starting state | on the infinite line left of tree. • Apply Hamiltonian for O(N) time. • If T=1, the new state is on the right (transmission). • If T=0, the new state is on the left (reflection). … … Proof: reflection coefficients of the tree.

22. Next steps [A, Childs, Reichardt, Špalek, Zhang, 2007]

23. AND OR OR AND AND x1 x2 x3 x4 x5 x7 x6 Improvement I Quantum algorithm for arbitraryformulas

24. Our result • query quantum algorithm for any size-N formula. Quantum speedups for anything that can be expressed by logic formulas

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

26. Loose end III • FGG algorithm uses scattering theory and looks very different from the previous quantum algorithms. • Our work: relations to search, amplitude amplification. • New understanding of FGG.

27. Two reflections • [Aharonov, 98] Analysis of Grover’s algorithm; • Other applications: • Amplitude amplification; • Quantization of Markov chains. • Now: logic formulas.

28. Beyond logic formulas [Reichardt, Špalek, 2008] • Input x1, ..., xN vectors v1, ..., vM. • Output F(x1, ..., xN) = 1 if there are vi1,vi2, ..., vik v=vi1+vi2+...+vik. Span program with witness size T O(√T) query quantum algorithm

29. Span programs [Reichardt, Špalek, 2008] Logic formula of size T Span program with witness size T O(√T) query quantum algorithm

30. Span programs [Reichardt, 2009] Span program with witness size T  O(√T) query quantum algorithm

31. Adversary bound [A, 2001, Hoyer, Lee, Špalek, 2007] • Boolean function f(x1, ..., xN); • Inputs x = (x1, ..., xN); • Theorem If there is a matrix A: A[x, y]≠0 only if f(x) ≠ f(y), then computing f requires quantum queries

32. Span programs [Reichardt, 2009] Optimal span program Semidefinite program (SDP) Dual SDP Optimal adversary bound

33. Big question #1 What other problems have quantum speedups?

34. Big question #2 What properties of a problem imply a quantum speedup?

35. Structural results • [Beals et al., 1998] Let f(x1, ..., xN) – total Boolean function. Then, D(f) ≤ Q6(f). • [Aaronson, A, 2008] Let f(x1, ..., xN) – symmetric function. Then, R(f) ≤ Q9(f). • R(f), Q(f) – number of variables that should be evaluated by classical/quantum algorithm.

36. Open question • Other classes of problems for which speedup is at most polynomial?