Quantum random walks and quantum algorithms

Andris AmbainisUniversity 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... ...

Continuous time version: move left/right at certain rate.

Cont. time quantum walk Random

walk:

Quantum walk:

......

...0

......

10

......

0......1

...0

01

10

0...

1......0

......

01

......

0...

......

A

Adjacency matrix:

Apdt

dp

iAdt

d

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... ...

Quantum walk on line

States |x, d, x –location, d-direction.

-2 -1 0 1 2... ...

rightleftright

rightleftleft

|2

1|

2

1|

|2

1|

2

1|

rightxrightx

leftxleftx

,1,

,1,

“Coin flip”:

Shift:

Classical vs. quantum

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

3.50E-01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Run for t steps, measure the final location.

Distance: (t) Distance: (t)

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 ...

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 ...

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.

0 1 2 ... 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

Explanationtime

location

A second boundaryreflects part of the state

Quantum walk on general graphs

H – adjacency matrix of a graph.

iHd

iHe

Discrete quantum walk

Discrete quantum walk Edges: |u, v.

1. “Coin flip”:

wuvuw

uw ,,

2. “Shift”:uvvu ,,

Part 2

Applications of quantum walks

Quantum search on grids [Benioff, 2000]

N* N grid. Each location stores a

value. Find a location

storing a certain value.

Grover’s search

Find i for which xi=1. Questions: ask i, get xi. Classically, N questions. Quantum, O(N) questions [Grover,

1996].

0 1 0 0...

x1 x2 xNx3

Quantum search on grids [Benioff, 2000]

Distance between opposite corners = 2N.

Grover’s algorithm takes

steps.

NNN

No quantum speedup.

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.

Quantum walk on grid

Basis states |x,y,, |x, y, , |x, y, , |x, y, .

Coin flip on direction:

2

1

2

1

2

1

2

12

1

2

1

2

1

2

12

1

2

1

2

1

2

12

1

2

1

2

1

2

1

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,

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.

log NNO

*Improved to const [Tulsi, 2008]

Element distinctness

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.

7 9 2 1...

x1 x2 xNx3

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}

{1,3}

{1,4}

{1, 2, 3}

{1, 2, 4}

N2/3 N2/3+1

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}

{1,3}

{1,4}

{1, 2, 3}

{1, 2, 4}

A quantum walk finds a marked vertex in N2/3 steps.

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.

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.

AND-OR tree

AND

OR OR

x11 x22 x33 x44

OR OR

x55 x66 x77 x88

AND

OR

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.

AND

OR OR

x11 x22 x33 x44

i

xi

ii iaia i)1(

Results Full binary tree of depth

d. N=2d leaves. Deterministic: (N). Randomized [SW,S]:

(N.753…). Quantum? Easy q. lower bound:

(N).

AND

OR OR

x11 x22 x33 x44

[Farhi, Goldstone, Gutmann]:

O(N) time quantum algorithm in Hamiltonian query model

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.

Improvement I

AND

OR OR

AND ORx11 x22

x33 x44 x55 x66

Quantum algorithm for unbalanced trees

Improvement II

O(N) time Hamiltonian quantum algorithm

O(N1/2+o(1)) query quantum algorithm

[Farhi, Goldstone, Gutmann]:

We can design discrete query algorithm directly.

[Childs et al.]:

…

Finite “tail” in one direction

0 1 1 0

[Childs et al.]:

…

Basis states |v, v – vertices of augmented tree.

Hamiltonian H, H-adjacency matrix of augmented tree.

[Childs et al.]:

…1-1-11

Starting state:

j

jstart j2)1(

Hamiltonian H,

H – adjacency matrix

What happens? If T=0, the state

stays almost unchanged.

If T=1, the state scatters into the tree.

…

0 1 1 0

Surprising: the behaviouronly depends on T, not x1, …, xN.

More precisely… 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… T=0: H has a

0-eigenstate. T=1: All eigenvalues

are at least 1/N.

…

0 1 1 0

Time 1/min eigenvalue O(N)

From Hamiltonians to unitaries

H0- AND-OR formula

H1 – extra edges for xi=1

H=H0+H1

U=U1 U0

From Hamiltonians to unitaries

…

U0|=-| if H0|=|, 0.

U1|v=-|v if v contains xi=1.

0-eigenstate of H 1-eigenstate of U1U0

Handling unbalanced trees Weighted adjacency matrix H:

Huv0 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.

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(S1+), Depth(F’)=O(log S).2. Size(F’)= , Depth(F’)=

SSO log/11 SO log2

O(N1/2+) quantum algorithm for any formula F

[Reichardt, Spalek]

MAJ

x11 x22 x33

MAJ

MAJ

MAJ

x44 x55 x66 x77 x88 x99

MAJORITY tree: O(2d), optimal.

Span programs

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?

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 propertyT=1 – global property