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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