quantum random walks – new method for designing quantum algorithms andris ambainis university of...
TRANSCRIPT
![Page 1: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/1.jpg)
Quantum random walks – new method for designing quantum algorithms
Andris AmbainisUniversity of Latvia
![Page 2: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/2.jpg)
Quantum computing New model of computing, based on
quantum mechanics. More powerful than conventional
(classical) computing.
![Page 3: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/3.jpg)
Talk outline
1. Main results of quantum computing.
2. The model.3. Quantum algorithms based on
quantum walks.
![Page 4: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/4.jpg)
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.
![Page 5: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/5.jpg)
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
![Page 6: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/6.jpg)
NP-complete problems Does this graph
have a Hamiltonian cycle?
Hamiltonian cycles are: Easy to verify; Hard to find (too
many possibilities).
![Page 7: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/7.jpg)
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
![Page 8: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/8.jpg)
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.
![Page 9: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/9.jpg)
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].
![Page 10: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/10.jpg)
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
![Page 11: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/11.jpg)
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”.
![Page 12: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/12.jpg)
One way to design quantum algorithms
Search procedure
Amplify quantumly:
Classical algorithm + amplitude amplification
![Page 13: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/13.jpg)
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.
![Page 14: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/14.jpg)
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
![Page 15: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/15.jpg)
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).
![Page 16: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/16.jpg)
The model of quantum computing
![Page 17: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/17.jpg)
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
![Page 18: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/18.jpg)
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.
![Page 19: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/19.jpg)
Probabilistic computation Pick the next
state, depending on the current one.
1
2 3
4
2/3
1/3
![Page 20: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/20.jpg)
Probabilistic computation
1
2 3
4
Transitions: rij - probabilities to move from i to j.
i
ijij rpp'2/3
1/3
![Page 21: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/21.jpg)
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
![Page 22: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/22.jpg)
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
![Page 23: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/23.jpg)
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
![Page 24: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/24.jpg)
Allowed transitions
U – unitary: If , then .
MMM
M
uu
uu
...
.........
...
1
111
M'
...
'1
M
...1
i
i 12
ii 1'2
![Page 25: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/25.jpg)
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
![Page 26: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/26.jpg)
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?
![Page 27: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/27.jpg)
Measurement
Quantum state:
1 1 + 2 2 + … + M M
|1|2
1
prob. |2|2
2
|M|2
M…
Measurement
![Page 28: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/28.jpg)
Quantum walks and quantum algorithms
1. Quantum random walks.2. Grover’s quantum search
algorithm.3. Quantum walks + quantum
algorithms.
![Page 29: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/29.jpg)
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... ...
![Page 30: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/30.jpg)
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... ...
![Page 31: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/31.jpg)
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:
![Page 32: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/32.jpg)
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 1 2... ...
Left:
Right:
![Page 33: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/33.jpg)
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 1 2... ...
Left:
Right:
![Page 34: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/34.jpg)
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 1 2... ...
Left:
Right:
![Page 35: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/35.jpg)
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0 2... ...
Left:
Right:
![Page 36: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/36.jpg)
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0... ...
Left:
Right:
1
![Page 37: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/37.jpg)
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0... ...
Left:
Right:
1
![Page 38: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/38.jpg)
Quantum walk on line
-2 -1 0 1 2... ...
-2 -1 0... ...
Left:
Right:
1
![Page 39: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/39.jpg)
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
![Page 40: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/40.jpg)
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)
![Page 41: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/41.jpg)
Grover’s quantum search algorithm
![Page 42: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/42.jpg)
Grover's search
Find i such that xi=1. Queries: ask i, get xi.
0 1 0 0...
x1 x2 xNx3
![Page 43: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/43.jpg)
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)
![Page 44: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/44.jpg)
Queries in the quantum world
12 3
…
N…
xi=1
12
…
…
![Page 45: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/45.jpg)
Grover’s algorithm times repeat:
Query; Diffusion transformation:
N4
NNN
NNN
NNN
21...
22............
2...
21
2
2...
221
![Page 46: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/46.jpg)
Analysis of Grover’s algorithm
For simplicity, assume exactly one i:xi=1.
N
1
… N
1
…
N
1
1st query
![Page 47: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/47.jpg)
Diffusion transformation
N
1
…
N
1
Inversion against average
N
1
…
N
1
N
3
D
![Page 48: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/48.jpg)
2nd query
N
1
…
N
1
N
3
N
1
…
N
1
N
3
N
3
![Page 49: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/49.jpg)
2nd diffusion
N
1
…
N
1
N
3
N
3
N
1
…
N
1
N
3
N
3
N
5
![Page 50: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/50.jpg)
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?
![Page 51: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/51.jpg)
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.
![Page 52: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/52.jpg)
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
![Page 53: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/53.jpg)
[Benioff, 2000] n* n grid. Distance between
opposite corners = 2n. Grover’s algorithm
takes
steps. No quantum speedup.
nnn
Quantum walks solve this problem!
![Page 54: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/54.jpg)
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.
![Page 55: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/55.jpg)
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
![Page 56: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/56.jpg)
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.
![Page 57: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/57.jpg)
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]
![Page 58: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/58.jpg)
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]
![Page 59: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/59.jpg)
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
![Page 60: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/60.jpg)
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}
![Page 61: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/61.jpg)
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.
![Page 62: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/62.jpg)
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
![Page 63: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/63.jpg)
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)
![Page 64: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/64.jpg)
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).
![Page 65: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/65.jpg)
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.
![Page 66: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/66.jpg)
Inside the Szegedy’s black box
How do we turn random walksinto quantum algorithms?
![Page 67: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/67.jpg)
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
![Page 68: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/68.jpg)
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.
![Page 69: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/69.jpg)
As in Grover’s search…
N
1
…
N
1
N
3
N
3
N
1
…
N
1
N
3
N
3
N
5
![Page 70: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/70.jpg)
Mathematics
Mp
p
...1 - vector of probabilities
1 step: P’ = MP
k steps: P’ = MkP
Eigenvectors and eigenvalues of M,M – transition matrix.
![Page 71: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/71.jpg)
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
![Page 72: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/72.jpg)
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 '
![Page 73: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/73.jpg)
Other applications of quantum walks
![Page 74: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/74.jpg)
“Glued trees” [Childs et al., 02]
Two full binary trees of depth d;
Randomly connect leaves of the two graphs.
![Page 75: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/75.jpg)
“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.
![Page 76: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/76.jpg)
“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).
![Page 77: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/77.jpg)
[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
![Page 78: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/78.jpg)
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.
![Page 79: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/79.jpg)
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…
![Page 80: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/80.jpg)
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
![Page 81: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/81.jpg)
NMR quantum computing
12 quantum bits (IQC): Creating a certain
quantum state. 7 quantum bits
(IBM): Factoring 15; Grover’s search
among 4 items.
![Page 82: Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia](https://reader035.vdocuments.us/reader035/viewer/2022062314/56649f1f5503460f94c37d27/html5/thumbnails/82.jpg)
Quantum cryptography Secure data
transmission, using quantum mechanics.
Only requires single quantum bits.
Implemented over 200km distance.
Available commercially.
QKD device ofId Quantique(Geneva)