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Quantum random walks – new method for designing quantum algorithms
Andris AmbainisUniversity of Latvia
Quantum computing New model of computing, based on
quantum mechanics. More powerful than conventional
(classical) computing.
Talk outline
1. Main results of quantum computing.
2. The model.3. Quantum algorithms based on
quantum walks.
Shor’s algorithm Factoring: given N=pq, find p and q. Best algorithm - 2O(n1/3), n – number of
digits. Quantum algorithm - O(n3) [Shor, 94]. Cryptosystems based on hardness of
factoring/discrete log become insecure.
Grover's search
Find i such that xi=1. Queries: ask i, get xi. Classically, N queries required. Quantum: O(N) queries [Grover, 96]. Speeds up any search problem.
0 1 0 0...
x1 x2 xNx3
NP-complete problems Does this graph
have a Hamiltonian cycle?
Hamiltonian cycles are: Easy to verify; Hard to find (too
many possibilities).
Quantum algorithm
Let N – number of possible Hamiltonian cycles.
Black box = algorithm that verifies if the ith candidate – Hamiltonian cycle.
Quantum algorithm with O(N) steps.
0 1 0 0...
x1 x2 xNx3
Applicable to any search problem
Pell’s equation Given d, find the smallest solution
(x, y) to x2-dy2=1. Probably harder than factoring and
discrete logarithm. Best classical algorithms:
for factoring; 2O(N) for discrete logarithm.
)( 3/1
2 NO
Hallgren, 2001: Quantum algorithm for Pell’s equation.
Number theory and algebraic problems Polynomial time quantum
algorithms: Factoring [Shor, 94] Discrete logarithm [Shor, 94]; Pell’s equation [Hallgren, 02]. Principal ideal problem [Hallgren, 02]. Computing the unit group [Hallgren,
05].
Grover's search
Find i such that xi=1. Queries: ask i, get xi. Classically, N queries required. Quantum: O(N) queries [Grover, 96]. Speeds up any search problem.
0 1 0 0...
x1 x2 xNx3
Amplitude amplification [Brassard et al., 01] Algorithm A that finds a certain
object with probability . How many repetitions to achieve
probability 2/3? Classically, (1/). Quantum: O(1/). “Quantum black box”.
One way to design quantum algorithms
Search procedure
Amplify quantumly:
Classical algorithm + amplitude amplification
Quantum counting
Determine the fraction of xi=1. E.g., distinguish whether the fraction is 1/2- or 1/2+.
Classical random sampling: O(1/2) steps. Quantum: O(1/) steps.
0 1 0 0...
x1 x2 xNx3
Can be used for more complicated statistics.
Element distinctness
Numbers x1, x2, ..., xN.
Determine if two of them are equal. Classically: N queries. Quantum: O(N2/3).
7 9 2 1...
x1 x2 xNx3
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).
The model of quantum computing
Probabilistic computation Probabilistic system
with finite state space.
Current state: probabilities pi to be in state i.
1
2 3
4
0.6
0.1 0.2
0.1
i
ip 1
Quantum computation Current state:
amplitudes i to be in state i.
1
2 3
4
0.4+0.3i
-0.7 0.4-0.1i
0.3
i
i 12
For most purposes, real (but negative) amplitudes suffice.
Probabilistic computation Pick the next
state, depending on the current one.
1
2 3
4
2/3
1/3
Probabilistic computation
1
2 3
4
Transitions: rij - probabilities to move from i to j.
i
ijij rpp'2/3
1/3
Probabilistic computation Probability vector (p1, …, pM). Transitions:
before the transition
MMM
M
rr
rr
...
.........
...
1
111
transition probabilities
Mp
p
'
...
'1
after the transition
Mp
p
...1
Allowed transitions
R –stochastic: If i pi = 1, then i p’i = 1.
MMMM
M
M p
p
rr
rr
p
p
...
...
.........
...
'
...
' 1
1
1111
Quantum computation Amplitude vector (1, …, M),
. Transitions:
before the transition
MMM
M
uu
uu
...
.........
...
1
111
transition matrix
M'
...
'1
after the transition
M
...1
i
i 12
Allowed transitions
U – unitary: If , then .
MMM
M
uu
uu
...
.........
...
1
111
M'
...
'1
M
...1
i
i 12
ii 1'2
Geometric interpretation (1, …, M),
- vectors on the unit sphere.
Transformations that preserve - rotations of the unit sphere.
i
i 12
i
i 12
1
0
Summary so far Quantum probabilistic with
complex probabilities. Instead of i pi = 1 we have
(l2 norm instead of l1). i
i 12
How do we go from quantum world to conventional world?
Measurement
Quantum state:
1 1 + 2 2 + … + M M
|1|2
1
prob. |2|2
2
|M|2
M…
Measurement
Quantum walks and quantum algorithms
1. Quantum random walks.2. Grover’s quantum search
algorithm.3. Quantum walks + quantum
algorithms.
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... ...
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... ...
rightxleftxrightx
rightxleftxleftx
,|2
1,|
2
1,|
,|2
1,|
2
1,|
rightxrightx
leftxleftx
,1,
,1,
“Coin flip”:
Shift:
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 1 2... ...
Left:
Right:
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 1 2... ...
Left:
Right:
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 1 2... ...
Left:
Right:
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 2... ...
Left:
Right:
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0... ...
Left:
Right:
1
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0... ...
Left:
Right:
1
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0... ...
Left:
Right:
1
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0... ...
Left:
Right:
1
-3
-3
3
2
1/21/8 1/8
1/8 1/8
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: (N) Distance: (N)
Grover’s quantum search algorithm
Grover's search
Find i such that xi=1. Queries: ask i, get xi.
0 1 0 0...
x1 x2 xNx3
Queries in the quantum world States |1, |2, …, |N. Query:
|i |i, if xi=0; |i -|i, if xi=1;
(a state with amplitude 1 at location i gets mappedto a state with amplitude –1 at i)
Queries in the quantum world
12 3
…
N…
xi=1
12
…
…
Grover’s algorithm times repeat:
Query; Diffusion transformation:
N4
NNN
NNN
NNN
21...
22............
2...
21
2
2...
221
Analysis of Grover’s algorithm
For simplicity, assume exactly one i:xi=1.
N
1
… N
1
…
N
1
1st query
Diffusion transformation
N
1
…
N
1
Inversion against average
N
1
…
N
1
N
3
D
2nd query
N
1
…
N
1
N
3
N
1
…
N
1
N
3
N
3
2nd diffusion
N
1
…
N
1
N
3
N
3
N
1
…
N
1
N
3
N
3
N
5
Grover: summary Query+diffusion increases the
amplitude of i:xi=1, decreases the amplitude of i:xi=0.
After repetitions the amplitude of i: xi=1 is almost 1.
Measuring at this point gives the solution i.
N4
Why N?
Probabilities vs. amplitudes
Probabilities: 1/N probability to
query the right i in 1 step.
N steps to obtain the probability 1.
Amplitudes: The amplitude of
the right i is 1/N. Each
query+diffusion increases it by 2/N.
After N steps it becomes 1.
Measurement: amplitude 1 probability 12=1.
Grid with N elements. Task: find the location
storing a certain value. In one step, we can
check the current location or move distance 1.
Search on grids
[Benioff, 2000] n* n grid. Distance between
opposite corners = 2n. Grover’s algorithm
takes
steps. No quantum speedup.
nnn
Quantum walks solve this problem!
Quantum walk search [Szegedy, 04]
Finite search space. Some elements might
be marked. Find a marked
element!
1
2 3
4 5
6 Perform a random
walk, stop after finding a marked element.
Conditions on Markov chain
Random walk must be symmetric: pxy=pyx.
Start state = uniformly random state.
T = expected time to reach marked state, if there is one.
1
2 3
4 5
6
Main result [Szegedy, 04]
Theorem Assume that:1. There are no marked states, or2. A marked state is reached in
expected time at most T.
A quantum algorithm can distinguish the two cases in time O(T).
Quadratic speedup for a variety of problems.
Application 1 N states. Is there a marked state? Random walk: at each
step move to a randomly chosen vertex.
Finds a marked vertex in N expected steps.
Quantum: O(N) steps[Grover]
Application 2: search on grids
Random walk: at each step move to a random neighbour.
Finding marked state: O(N log N) steps.
Quantum algorithm:
[A, Kempe, Rivosh, 2005]
Application 3: element distinctness
Numbers x1, x2, ..., xN.
Determine if two of them are equal. Well studied problem in classical CS. Classically: N steps.
7 9 2 1...
x1 x2 xNx3
Johnson graph Vertices: sets S, |S|
=k, S{1, 2, …, N}. Edges: (S, T), for S, T
that differ only in 1 element.
{1, 2, 3}
{1, 2, 4} {1, 3, 5}
{2, 3, 4} {3, 4, 5}
Random walk Marked vertices =
those for which S contains i, j: xi = xj.
Random walk in which we maintain xi, iS.
{1, 2, 3}
{1, 2, 4} {1, 3, 5}
{2, 3, 4} {3, 4, 5} K queries to start; 1 query to move to
adjacent vertex.
Search by random walk Time to find a marked
vertex: K queries to start; moving
steps.
{1, 2, 3}
{1, 2, 4} {1, 3, 5}
{2, 3, 4} {3, 4, 5}
K
NO
2
Quantum:
k
NO
Quantum algorithm Quantum time:
K queries to start; moving
steps.
Total:
{1, 2, 3}
{1, 2, 4} {1, 3, 5}
{2, 3, 4} {3, 4, 5}
k
NO
k
NkO
K=N2/3 O(N2/3)
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).
Matrix multiplication [Buhrman, Špalek, 05] A, B, C – n*n matrices. Finding C=AB: O(n2.37…) steps; 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.
Inside the Szegedy’s black box
How do we turn random walksinto quantum algorithms?
Quantum walk State space with states |i. Starting state
Quantum walk with two transition rules: “usual” for unmarked
vertices; “special” for marked.
1
2 3
4 5
6
M
i
iM 1
1
Quantum walk algorithm No marked
vertices: The starting state
is unchanged.
Some marked vertices: The starting state is
changed at the marked vertices.
Changes spread and accumulate.
Run for sufficiently many steps, test if the system is still in the starting state.
As in Grover’s search…
N
1
…
N
1
N
3
N
3
N
1
…
N
1
N
3
N
3
N
5
Mathematics
Mp
p
...1 - vector of probabilities
1 step: P’ = MP
k steps: P’ = MkP
Eigenvectors and eigenvalues of M,M – transition matrix.
Mathematics II
Eigenvectors of M: v1, …, vN.
M vi= i vi
Mt vi= (i)t vi
Express starting distribution via eigenvectors:
i
iivP i
it
iit vPM
Mathematics III Random walk
Probability matrix M;
Expected time T to hit a marked vertex.
Quantum walk Unitary matrix U; Time T’ at which the
state becomes sufficiently different from the starting state.
TOT '
Other applications of quantum walks
“Glued trees” [Childs et al., 02]
Two full binary trees of depth d;
Randomly connect leaves of the two graphs.
“Glued trees” [Childs et al., 02]
Start position: the left root.
Task: find the other root.
Vertices and edges not labeled.
2d+2-2 vertices.
Any classical algorithm takes (cd) steps.
“Glued trees” [Childs et al., 02]
Perform a quantum walk, starting from the left root.
After O(d2) steps, a quantum walk is at the right root.
Exponential speedup: O(d2) vs (cd).
[A, Childs, et al., 07] AND-OR formula of size M. Variables accessed by
queries: ask i, get xi. Theorem Any formula can
be evaluated with O(M1/2+o(1)) queries.
AND
OR OR
x11 x22 x33 x44
Speedup for anything that canbe expressed as a formula
Conclusion Quantum walks can be used to
design quantum algorithms for many problems.
Quantum walk search: create a classical random walk, quantize it with a speedup.
“Quantum black box” within a classical algorithm.
Building a quantum computer Classical computer operates on bits (0
or 1). Quantum bit: a system with basis
states |0, |1, general state 0|0+ 1|1.
Need physical systems that act like quantum bits…
10s of candidates…
NMR quantum computing (IBM, Waterloo, etc.)
Quantum computer = molecule.
Quantum bits = nuclear spins.
Operations performed using magnetic fields.
Spin – property that determines how the particle behave in a magnetic field
0, 1
NMR quantum computing
12 quantum bits (IQC): Creating a certain
quantum state. 7 quantum bits
(IBM): Factoring 15; Grover’s search
among 4 items.
Quantum cryptography Secure data
transmission, using quantum mechanics.
Only requires single quantum bits.
Implemented over 200km distance.
Available commercially.
QKD device ofId Quantique(Geneva)
