A quantum walk based search algorithm, and its optical realisation

A quantum walk based search algorithm,
and its optical realisation
Aurél Gábris
FJFI, Czech Technical University in Prague
with
Tamás Kiss and Igor Jex
Prague, Budapest
Student Colloquium and School
on Mathematical Physics
Stará Lesná, Slovakia
23 – 29th August 2008
1 / 26

Outline
Quantum walks on graphs
The quantum mechanical search problem
The SKW quantum walk search algorithm
Optical implementation
Analysis of errors
J. Kempe: Contemp. Phys. 44 307 (2003), quant-ph/0303081
N. Shenvi, J. Kempe and K. B. Whaley: Phys. Rev. A 67, 052307 (2003)
A. Gábris, T. Kiss and I. Jex: Phys. Rev. A 76, 062315 (2007)
2 / 26

Quantum walks on graphs
Graph G = (V, E)
state of system: |ψ〉 ∈ H V = Span{|v i 〉 : v i ∈ V}
dynamics: according to E
How to go from classical to quantum walks
classical
prob. distrib.
stochastic
Two alternative approaches
discrete time quantum walk
continuous time quantum walk
quantum
quantum state
unitary
(relationship: Strauch: Phys. Rev. A 74, 030301 (2006)
Introductory review: J. Kempe: Contemp. Phys. 44, 307 (2003),
quant-ph/0303081
3 / 26

Discrete time quantum walk (coined QW)
Focussing generalization on:
Stochastic process → Unitary process
Simple example
Galton’s board (quincunx)
at each pin: 50%-50% probability
→ tossing a coin
Resolution
Replace classical coin by quantum coin!
{|j〉 |v j 〉 : j = 1, . . . , degree(v j )}
additional degree of freedom
4 / 26

Discrete quantum walk
G = (V, E) n-regular graph
coin states: 1, . . . , n ⇒ H C = Span {|j〉 | j = 1, . . . , n}
H = H C ⊗ H V
Dynamics
coin operator: C ∈ Lin(H C ), unitary
translation operator: S ∈ Lin(H), unitary
step operator: U = S(C ⊗ 1), uniform coin
Coin operators
∣ ∣
balanced: ∣Cij ∣∣ 2
= 1/d
unbalanced
permutation invariant
Grover coin: G = |s〉〈s| − 1
Grover (diffusion) operator
⎛
2
d − 1 2 2
d d
· · ·
2 2
d d − 1 2
d
· · ·
2 2 2
d d d
⎜⎝
− 1
.
. .
..
⎞⎟⎠
5 / 26

Continuous time quantum walk
Discrete transition probabilities
p t = (p t 1 , pt 2 , . . . , pt |V| )
p t+1 = Mp t
M ij transition prob.
Continuous time Markov chain (classical)
⎧
−γ, if i j, {v i , v j } ∈ E
dp(t)
⎪⎨
= −Hp(t) H ij = 0, if i j, {v
dt
i , v j } E
⎪⎩ d i γ, if i = j (d i : rank of v i )
⇒ p(t) = e −Ht p(0)
Quantum mechanical generalization
∑
Choose: Ĥ = γ d i |v i 〉〈v i | − γ
i
⇒ Û(t) = e −iĤt
∑
{v i ,v j }∈E
|v i 〉〈v j |
6 / 26

Applications of QWs
Designing efficient quantum algorithms is difficult
QWs may be used for this task
Some findings:
traversal of binary trees (exp. faster hitting time)
traversal of randomly connected binary trees
(quadratic algorithmic speedup)
evaluation of NAND formulas
quantum searching
. . .
7 / 26

The quantum mechanical search problem
Classical specification
given search space X
given function f : X → {0, 1}
task: find ∀x ∈ X s. t. f (x) = 1
Quantum mechanics
Try O: O |x〉 = |f (x)〉 ⇒ not unitary!
Unitary choice:
i.e. O |x〉 |0〉 = |x〉 |f (x)〉
O |x〉 |s〉 = |x〉 |f (x) ⊕ s〉
Complexity
O oracle operator
Often measured as the number of oracle calls (query complexity)
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Quantum walk on a hypercube
hypercube in n dimensions:
n-regular graph
coined quantum walk
(n = 4)
Vertices
n dimensions → x ∈ {0, 1, . . . , 2 n − 1} integer
binary string repr.: ⃗x, n binary digits
Useful definitions
Hamming weight: |⃗x| sum of digits/number of 1s
Hamming distance: d(⃗x,⃗y) = |⃗x ⊕ ⃗y|
Edges
⃗x,⃗y ∈ V are connected iff d(⃗x,⃗y) = 1
9 / 26

Grover walk on a hypercube
Definition
Graph: n dimensional hypercube
n-regular graph
N = 2 n vertices, → labels: n bit long binary strings
(⃗x = 0, . . . , 2 n − 1) (9 = 0 . . . 01001)
Coin: C = G ⊗ 1 (Grover operator)
∑
Propagator: S = |d,⃗x ⊕ ⃗e d 〉〈d,⃗x|
Symmetries
⃗x,d
(
⃗ed = 2 d , edges )
“shift” symmetry: ⃗x → ⃗x ⊕ ⃗x s
permutation symmetry: P ij swap bits at positions i, j
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Spectrum of Grover walk on a hypercube
Solve problem in Fourier space. . .
“Trivial” eigenvectors
λ = ±1
“Non-trivial” eigenvectors
λ | ⃗k|
= exp(±iω | ⃗k| ) = 1 − 2| ⃗k|
n
|v ⃗ k 〉 , |v∗ ⃗k 〉 = ∑
⃗x,d
(−1) ⃗ k⃗x 2−n/2
√
± 2i
n
|⃗k|(n − |⃗k|)
⎧
⎪⎨ 1/ √ k, if k d = 1
√ |d,⃗x〉
2
⎪⎩ ∓i/ √ n − k, if k d = 0
11 / 26

The SKW algorithm
SKW quantum walk (perturbed Grover walk)
1 ∑
C
initial state: √ |d,⃗x〉 ,
0 = G,
nN C 1 = −1
d,⃗x
O marks |⃗x tg 〉 =⇒ C ′ = C 0 ⊗ 1 + (C 1 − C 0 ) ⊗ |⃗x tg 〉〈⃗x tg |
time step operator: U ′ = SC ′
After optimal number of steps: t f = O( √ N)
⇒ probability of |⃗x tg 〉: p 0 = ∑ | 〈d,⃗x tg |ψ(t f )〉 | 2 = 1 2 − O ( )
1
n
d
Classical analysis and protocol
1 Execute SKW QW
2 obtain ⃗x m
3 verify ⃗x m using O
−→ repeat until ⃗x tg found: 1 − ε certainty → r ε repetitions
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Proof of principle (sketch)
Collapse onto a line
⃗x tg can be chosen ⃗x tg = ⃗0:
⇒ C = C 0 ⊗ +(C 1 − C 0 ) ⊗ |⃗0〉〈⃗0|
C 0 , C 1 permutation symmetric
|⃗0〉 permutation symmetric
|ψ 0 〉 permutation symmetric
⇒ time evolution in an invariant subspace
Basis: |R, x〉, |L,
∑
x〉 (x
∑
= 0, 1, . . . , n)
∑
|R, x〉 = N R,x |d,⃗x〉, |L, x〉 = N L,x
|⃗x|=x x d =0
Re-formulate the QW
|ψ 0 〉 : express in |R, x〉, |L, x〉 basis
U ′ = U − 2 |L, 1〉〈R, 0|
prob. success: p 0 = | 〈R, 0|ψ f 〉 | 2
∑
|d,⃗x〉
|⃗x|=x x d =1
|ψ 1 〉 has property such that, | 〈R, 0|ψ 1 〉 | 2 = 1/2.
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Proof of principle (sketch)
Collapse onto a line
⃗x tg can be chosen ⃗x tg = ⃗0:
100
110
⇒ C = C 0 ⊗ +(C|d,⃗x〉
1 − C 0 ) ⊗ |⃗0〉〈⃗0|
1/3
C 0 , C 1 permutation symmetric
010
101
000
111
|⃗0〉 1/3permutation symmetric
1/3
|ψ 0 〉 permutation symmetric
001 110
⇒ time evolution in an invariant subspace
Basis:
1
|R,
2/3
x〉,
1/3
|L,
∑
x〉 (x
∑
= 0, 1, . . . , n)
|L, x〉, |R, x〉
∑ ∑
|R, x〉 = N R,x |d,⃗x〉, |L, x〉 = N L,x |d,⃗x〉
0
1 2 3
1/3 2/3
1
|⃗x|=x x d =0
Re-formulate the QW
|ψ 0 〉 : express in |R, x〉, |L, x〉 basis
U ′ = U − 2 |L, 1〉〈R, 0|
prob. success: p 0 = | 〈R, 0|ψ f 〉 | 2
|⃗x|=x x d =1
|ψ 1 〉 has property such that, | 〈R, 0|ψ 1 〉 | 2 = 1/2.
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Proof of principle (sketch) 2
Treat as perturbation
U ′ = U + ∆U
permutation symmetric
subspace:
no trivial eigenvalues
eigenvectors:
|v k 〉 = ( 1 ∑
n
k)
|⃗k|=k
Perturbed eigenvalues
QUANTUM RANDOM-WALK SEARCH ALGORITHM
|v ⃗ k 〉
ω ′ 0 ≈ 1
√
2 n−1
PHYSICAL
Corresponding eigenvectors
|ψ 0 〉 ≈ 1 √
2
(|ω ′ 0 〉 + |−ω′ 0 〉)
|ψ 1 〉 ≈ 1 √
2
(|ω ′ 0 〉 − |−ω′ 0 〉) =⇒ (U ′ ) t |ψ 0 〉 ≈ cos ω ′ 0 t |ψ 0〉 − sin ω ′ 0 t |ψ 1〉
Optimal time
t f =
π
2|ω ′ 0 | = π 2
√
2 n−1 + O ( )
1
n
14 / 26

Linear quantum optics
Basic elements
beam splitter
described by an SU(2) matrix R
(â out , ˆb out ) = R (â in , ˆb in )
b out
R
phase shifter
changes optical path length
â out = e iϕ â in
a in
b in
a out
Optical multiport
n input, n output
transformation: SU(n)
beam splitter: 2–2 multiport
Can be assembled from beam splitters (and phase shiters)
15 / 26

Quantum optical realization of SKW algorithm
n − n multiport
0
1
n − 1
1
n 0
n
n − 1
Single photon
|d〉 ≡ one photon at port d
transformation:
n−1 ∑
d=0
a d |d〉 → n−1 ∑
→ coin operator
d,k=0
C dk a k |d〉
16 / 26

Quantum optical realization of SKW algorithm
n − n multiport
0
1
n
n − 1
Scattering random walk
x = 0
t
t + 1
n − 1
1
n 0
transformation:
n−1 ∑
d=0
a d |d〉 → n−1 ∑
→ coin operator
d,k=0
C dk a k |d〉
x = 1
x = 2
x = 3
x = 4
|d, x〉 → |d, x ⊕ e d 〉
16 / 26

Quantum optical realization of SKW algorithm
n − n multiport
0
1
n
n − 1
Scattering random walk
x = 0
t
t + 1
n − 1
1
n 0
transformation:
n−1 ∑
d=0
a d |d〉 → n−1 ∑
→ coin operator
d,k=0
C dk a k |d〉
x = 1
x = 2
x = 3
x = 4
|d, x〉 → |d, x ⊕ e d 〉
S(C 0 ⊗ 1) ∑ a dx |d, x〉 = ∑ C dk a kx |d, x ⊕ e d 〉
dx
dxk
16 / 26

Quantum optical realization of SKW algorithm
n − n multiport
0
1
n
n − 1
n −With 1 loopback 1 x = 1
Ansingle column: scattering 0 quantum random walk
(similar: by Bužek and Košík) x = 2
transformation:
n−1 ∑
d=0
a d |d〉 → n−1 ∑
→ coin operator
d,k=0
C dk a k |d〉
Scattering random walk
x = 0
x = 3
x = 4
S(C 0 ⊗ 1) ∑ a dx |d, x〉 = ∑ C dk a kx |d, x ⊕ e d 〉
dx
dxk
t
t + 1
|d, x〉 → |d, x ⊕ e d 〉
16 / 26

QRW search in linear optical network
Column of multiports
x = 0
x = 1
x = 2
x = 3
x = 4
Consequence
Coin at x = 1 differs!
non-uniform coin → QRW search
17 / 26

QRW search in linear optical network
Column of multiports
x = 0
x = 1
x = 2
x = 3
x = 4
Consequence
Coin at x = 1 differs!
non-uniform coin → QRW search
Graph topology: hypercube
connect:
port d of x → port d of x ⊕ 2 d 17 / 26

QRW search in linear optical network
Column of multiports
x = 0
x = 1
x = 2
x = 3
x = 4
Consequence
Coin at x = 1 differs!
non-uniform coin → QRW search
Graph topology: hypercube
connect:
port d of x → port d of x ⊕ 2 d
Search problem
Find the multiport different from the
rest!
17 / 26

Losses, errors and decoherence
Decoherence
Basis of quantum computing:
“quantum coherence”
Losses and errors are inevitable in all physical realization:
non-unitary time evolution
Typical errors in optics
Photon loss (absorption or scattering in media)
Phase errors (difference in optical paths)
18 / 26

Uniform loss
Loss model
D(ϱ) = η 2 ϱ + (1 − η 2 ) |0〉〈0|
after t iterations:
ϱ → D t (U t ϱU †t )
Initial pure state |ψ〉: (η ≤ 1)
|ψ〉 → D |ψ〉 = η |ψ〉
Evolution operator: U ′′ = ηU ′ 19 / 26

Uniform loss
Loss model
D(ϱ) = η 2 ϱ + (1 − η 2 ) |0〉〈0|
after t iterations:
ϱ → D t (U t ϱU †t )
Initial pure state |ψ〉: (η ≤ 1)
|ψ〉 → D |ψ〉 = η |ψ〉
Evolution operator: U ′′ = ηU ′ log 2 x ≈ −ε + n/2 − 1/2
Success probability
Maximum success probability (p max )
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-15
-10
-5
0
5
10
x = − log 2 η √ 2 n−1
η = 1 − 2 −ε
p max (η) =
for large n: p max (x) ≈ 1 2
[
exp(−2x acot x) 1
1+x 2 2 − O ( ( )]
1
n)
+ x acot x O
1
n
exp(−2x acot x)
1
1+x 2 19 / 26

Comparison with numerical results
Maximum success probability (p max )
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
2
4
ε = 12
ε = 8
ε = 6
ε = 2
6 8 10
Rank of hypercube (n)
12
14
p max (η) =
[
exp(−2x acot x) 1
1+x 2 2 − O ( ( )]
1
n)
+ a acot xO
1
n
20 / 26

Direction dependent loss
η 0
η 1
η n−1
η 0 , η 1 , . . . , η n−1 can be different
Warning: original symmetry of hypercube broken!
Walk cannot be collapsed onto a line
21 / 26

Direction dependent loss
Description
starting with pure state: D = n−1 ∑
d=0
η d |d〉〈d|, {η} = {η d | d = 0, . . . , n − 1}
a lower bound: p({η}, t) ≥ ¯η 2t { √
pi (t) − (η max /¯η) [ (η max /¯η) t − 1 ] } 2
¯η = η max + η min
2
21 / 26

Direction dependent loss
Description
starting with pure state: D = n−1 ∑
d=0
η d |d〉〈d|, {η} = {η d | d = 0, . . . , n − 1}
a lower bound: p({η}, t) ≥ ¯η 2t { √
pi (t) − (η max /¯η) [ (η max /¯η) t − 1 ] } 2
〈η〉 = 1 n
n−1 ∑
d=0
¯η = η max + η min
2
η d , δ d = η d − 〈η〉 , Q 2 = 1 n
n−1 ∑
d=0
δ 2 d , 21 / 26

Direction dependent loss
Description
starting with pure state: D = n−1 ∑
d=0
η d |d〉〈d|, {η} = {η d | d = 0, . . . , n − 1}
a lower bound: p({η}, t) ≥ ¯η 2t { √
pi (t) − (η max /¯η) [ (η max /¯η) t − 1 ] } 2
¯η = η max + η min
2
〈η〉 = 1 n
n−1 ∑
d=0
η d , δ d = η d − 〈η〉 , Q 2 = 1 n
n−1 ∑
d=0
δ 2 d ,
W3 = 1 n−1 ∑
n
δ 3 d
d=0
Taylor expansion
around 〈η〉, (using permutation symmetry)
p max ({η}) = p max (〈η〉) + BQ 2 + CW 3 + R
21 / 26

Numerical results
Second order Taylor coefficient (n = 8)
Second order Taylor coefficient (B)
(log 2 scale)
64
16
4
1
2 −2
2 −4
2 −6
2 −8
2 −10
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Average transmission (〈η〉)
0.9
1
Effect of higher orders: p max ({η}) ≥ p max (〈η〉) + 2 −n Q 2 22 / 26

Numerical results
Second order Taylor coefficient (n = 8)
Second order Taylor coefficient (B)
(log 2 scale)
64
16
Empirical lower bound
4
using size, and the elementary statistical properties of noise:
1
2 −2
2 −4
2 −6
2 −8
2 −10
0.1
〈η〉 and Q
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Average transmission (〈η〉)
Effect of higher orders: p max ({η}) ≥ p max (〈η〉) + 2 −n Q 2
0.9
1
22 / 26

Numerical results
Absolute improvement (n = 7)
40
Absolute improvement (%)
30
20
10
0
-10
-20
0
0.05
0.1
0.15 0.2 0.25 0.3
Second moment (Q)
0.35
0.4
0.45
p max ({η}) − p max (η max )
p max (η max )
increase efficiency by loss!
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Phase Errors
Description
F({ϕ}) = ∑ d,x
e iϕ dx
|d, x〉〈d, x|
U({ϕ}) = U ′ F({ϕ})
Stable phase in one run
|ψ t ({ϕ})〉 = U({ϕ}) t |ψ 0 〉
ϱ out = |ψ t ({ϕ})〉〈ψ t ({ϕ})| {ϕ}
Gaussian: 〈ϕ〉 = 0, ∆ϕ variance
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Phase Errors
Description
F({ϕ}) = ∑ e iϕ dx
|d, x〉〈d, x|
d,x
U({ϕ}) = U ′ F({ϕ})
Single runs
0.4
0.35
Average probability (n = 6)
Average probability of success
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
100
200 300
Number of steps (t)
∆ϕ = 3 o
∆ϕ = 6 o
∆ϕ = 9 o
∆ϕ = 12 o
400
500
Probability of success
0.3
0.25
0.2
0.15
0.1
0.05
0
0
20
40
60 80 100
Number of steps (t)
120
140
160
Stable phase in one run
|ψ t ({ϕ})〉 = U({ϕ}) t |ψ 0 〉
ϱ out = |ψ t ({ϕ})〉〈ψ t ({ϕ})| {ϕ}
Gaussian: 〈ϕ〉 = 0, ∆ϕ variance
24 / 26

Phase Errors
0.4
Description
0.35
F({ϕ}) = ∑ e iϕ dx
|d, x〉〈d, x|
0.3 d,x
U({ϕ}) = 0.25 U ′ F({ϕ})
0.2
Single runs
0.15
Probability of success
Average probability (n = 6)
Average probability of success
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
20
0.1
0.05
0
0
100
40 60 80 100
Number of steps (t)
120
140
160
Average probability (n = 6)
∆ϕ = 3 o
0.4
∆ϕ = 6 o
0.35
∆ϕ = 9 o
0.3
∆ϕ = 12 o
Average probability of success
0.25
0.2
0.15
0.1
0.05
0
0
100
200 300
Number of steps (t)
Stable phase in one run
|ψ t ({ϕ})〉 = U({ϕ}) t |ψ 0 〉
200 300 400
ϱ out = |ψ t ({ϕ})〉〈ψ t ({ϕ})| {ϕ}
Number of steps (t)
∆ϕ = 3 o
∆ϕ = 6 o
∆ϕ = 9 o
∆ϕ = 12 o
400
500
Gaussian: 〈ϕ〉 = 0, ∆ϕ variance
500
24 / 26

Phase Errors
Numerical results
Max. average success probability
0.25
0.2
0.15
0.1
0.05
0
4 5
6 7 8
rank of hypercube (n)
ideal
∆ϕ = 3 o
∆ϕ = 6 o
∆ϕ = 15 o
9
10
25 / 26

Conclusions
Quantum walks
Interesting for
new quantum algorithms
its mathematics
Quantum walk search algorithm
proposal for optical realization
analysis of errors and losses
J. Kempe: Contemp. Phys. 44 307 (2003), quant-ph/0303081
N. Shenvi, J. Kempe and K. B. Whaley: Phys. Rev. A 67, 052307 (2003)
A. Gábris, T. Kiss and I. Jex: Phys. Rev. A 76, 062315 (2007)
26 / 26