Quantum random walks and quantum algorithms Andris Ambainis

Quantum random walks and quantum algorithms Andris Ambainis University of Latvia

Part 1 Quantum walks as a mathematical object

Random walk on line. . . n n -2 -1 0 1 2 . . . Start at location 0. At each step, move left with probability ½, right with probability ½. Continuous time version: move left/right at certain rate.

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

Random walk on line. . . n n -2 -1 0 1 2 . . . State (x, d), x –location, d-direction. At each step, n n n Let d=left with prob. ½, d=right w. prob. ½. (x, left) => (x-1, left); (x, right) => (x+1, right).

Quantum walk on line. . . n -2 -1 0 1 2 . . . States |x, d , x –location, d-direction. “Coin flip”: Shift:

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

Semi-infinite walk 0 n n 1 2 . . . 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.

Semi-infinite walk [A, Bach, et al. , 01] 0 n n 1 2 . . . What is the probability of stopping? Classically, 1. Quantumly, 2/. With some probability, quantum walk “never reaches” – 1.

Finite walk [Bach, Coppersmith, et al. , 2003] 0 n n 1 2 . . . n 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. Surprising, for two reasons

Probabilities to stop at -1 Classical Quantum Boundaries at – 1 and n n/(n+1) 1/ 2, for large n Semi-infinite 1 2/ “Semi-infinite” is not limit of “large n” 1/ 2 > 2/ Having a faraway border increases the chance of returning to -1

Explanation time A second boundary reflects part of the state location

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

Discrete quantum walk

Discrete quantum walk n Edges: |u, v. 1. “Coin flip”: 2. “Shift”:

Part 2 Applications of quantum walks

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

Grover’s search 0 1 0. . . 0 x 1 x 2 x 3 x. N n n Find i for which xi=1. Questions: ask i, get xi. Classically, N questions. Quantum, O( N) questions [Grover, 1996].

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

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

Quantum walk on grid n n Basis states |x, y, , |x, y, . Coin flip on direction:

Quantum walk on grid n Shift: n n |x, y, |x-1, y, |x+1, y, |x, y-1, |x, y+1,

Search by quantum walk n Perform a quantum walk with “coin flip”: n n n 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]

Element distinctness 7 9 2. . . 1 x 2 x 3 x. N n n n 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.

Element distinctness as search on a graph {1, 2} n {1, 2, 3} n {1, 3} {1, 2, 4} n {1, 4} n 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, xi=xj. In one step, we can n N 2/3+1 n Check if vertex marked; or Move to adjacent vertex.

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

Hitting times n n 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.

Testing matrix multiplication [Buhrman, Spalek 03] n n 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.

AND-OR tree OR AND OR x 1 OR x 2 x 3 OR x 4 x 5 OR x 6 x 7 x 8

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

Results n n AND n n OR x 1 OR x 2 x 3 x 4 n n Full binary tree of depth d. N=2 d leaves. Deterministic: (N). Randomized [SW, S]: (N. 753…). Quantum? Easy q. lower bound: ( N).

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

Flurry of improvements n n n A. Childs, B. Reichardt, R. Spalek, S. Zhang. ar. Xiv: quant-ph/0703015. A. Ambainis, ar. Xiv: 0704. 3628. B. Reichardt, R. Spalek, ar. Xiv: quantph/0710. 2630.

Improvement I AND OR OR x 1 x 2 AND x 3 OR x 4 x 5 x 6 Quantum algorithm for unbalanced trees

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

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

[Childs et al. ]: n n … Basis states |v , v – vertices of augmented tree. Hamiltonian H, Hadjacency matrix of augmented tree.

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

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

More precisely… n 0 1 1 0 n … 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.

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

From Hamiltonians to unitaries H 1 – extra edges for xi=1 H=H 0+H 1 U=U 1 U 0 H 0 - AND-OR formula

From Hamiltonians to unitaries U 0| =-| if H 0| = | , 0. … U 1|v =-|v if v contains xi=1. 0 -eigenstate of H 1 -eigenstate of U 1 U 0

Handling unbalanced trees Weighted adjacency matrix H: n n n 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: U 0| =-| if H| = | , 0.

Results (general trees) n n Theorem Any AND-OR formula of depth d can be evaluated with O( Nd) queries. BCE 91: 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

[Reichardt, Spalek] MAJ x 1 x 2 MAJ x 3 x 4 x 5 MAJ x 6 x 7 MAJORITY tree: O(2 d), optimal. Span programs x 8 x 9

Summary: applications n Quantum walks allow to solve: n n n 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?

Search vs. n n If no marked states, quantum walk stays in the start state. Otherwise, walk moves to marked states. formulas n n 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