quantum walk based search algorithms · quantum walks [meyer’96, watrous’98, aakv’01...

Quantum walk based search algorithms

Miklos Santha

CNRS, LRI, Universite Paris-Sud

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Overview of the talk

1 Classical search algorithms

2 Quantization of Markov chains

3 Quantum search algorithms

4 Applications

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Classical search

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Abstract search problem

Unordered SearchInput: Set X with M ⊆ X of marked elements, where ε = |M|/|X |Output: An element x ∈ M (or decide if there is any).

Search Algorithm 1 (Naive)Repeat for O(1/ε) steps

1 Pick uniformly random x ∈ X .

2 Check if x ∈ M, output if it is.

Additional structure for generating samples: Markov chain P on X .

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Markov chains

Random walks on directed graphs: P = (pxy )

Abstract search problem

The problem

Input:

! a set of elements X

! with unknown subset of

marked elements M ! X“! = |M|

|X |”

Output:

! a marked element x " M

t

y z

x

pxy pxz

m

n

Available procedures

! Setup (cost S):

pick a random x " X! Check (cost C):

check whether x " M! Update (cost U):

make a random walk P

x

y z

pxy pxz

P =

! here: assume P ergodic, symmetric

! " = e-v gap of P

• Irreducible: Strongly connected=⇒ unique stationary distribution π = (πx)

• Ergodic: Also aperiodic=⇒ eigenvalue gap δ = 1− 2nd largest |eigenvalue|

• Reversible: πxpxy = πypyx

Undirected graph with weighted edges

• Symetric =⇒ π is the uniform distributionUndirected regular graph

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Associated costs (symmetric chains)

Maintain some database d on X . The data associated with x isd(x).

• Setup cost S: Pick uniformly random x ∈ X , andconstruct d(x).

• Update cost U: Make one step from x to y according to P,and to update d(x) to d(y).

• Checking cost C: Check if x ∈ M using d(x).

Cost of Search Algorithm 1: (S + C)/ε,

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Two Markov chain based search algorithmsIdea: Replace random sampling by the walk

Search Algorithm 21 Pick random x ∈ X .

2 Repeat O(1/ε)-times

1 Check if x ∈ M.2 Make O(1/δ) steps of the walk P starting with x .

Cost: S + (U/δ + C)/ε

Search Algorithm 31 Pick random x ∈ X .

2 Repeat O(1/εδ)-times

1 Check if x ∈ M.2 Make one step of P starting with x .

Cost: S + (U + C)/δε7/30

Random walk on edgesWant to simulate a walk on vertices by a walk on edges:

P irreducible Markov chain Pedge walk on directed edges

P : |u〉 −→ |x〉 pxy−→ |y〉 pyz−→ |z〉Idea:

P at |x〉 and comes from |u〉 Pedge at |x〉|u〉|x〉 pxy−→ |y〉 |x〉|u〉 pxy−→ |y〉|x〉

Definition Pedge = SPright, where

• Controlled stochastic right flip Pright : |x〉|u〉 pxy−→ |x〉|y〉• Shift S : |x〉|y〉 1−→ |y〉|x〉

2 steps of of the edge walk are PedgePedge = SPrightSPright:

|x〉|u〉 pxy−→ |y〉|x〉 pyz−→ |z〉|y〉This is the same as PleftPright, got rid of the shift!

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Quantization of Markov chains

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Quantum walks [Meyer’96, Watrous’98, AAKV’01 Ambainis’04]

Classical walk on vertex space X does not quantizeIdea: Take walk space as X × C where C is some coin spaceDefinition W = SF , product of a quantum coin flip F and a shift S

• d-regular graphs: C = {1, . . . , d}• Quantum coin flip: Independent from vertex x

F |x〉|i〉 = |x〉D|i〉,where D is reflection about 1√

d

∑j |j〉 ( = Grover diffusion)

• Shift:S |x〉|i〉 = |i th neighbor of x〉|i〉

• general walk: C = X , walk on directed edges• Controlled quantum right flip:

Fright|x〉|y〉 = |x〉F x |y〉

• Shift: S |x〉|y〉 = |y〉|x〉Again, two steps of the walk is FleftFright

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Spectral characterization of W (P) on the walk subspace

Theorem [Szegedy’04]: P ergodic, reversible Markov chain withstationary distribution π. Let the eigenvalues of P be:

1 = cos θ0 > cos θ1 ≥ . . . ≥ cos θk > −1Then, on the walk space, the spectrum of W (P) is:

• |π〉 =∑

x∈X

√πx |x〉|px〉 is the unique 1-eigenvector,

• e±2iθj are eigenvalues for 1 ≤ j ≤ k ,

• All remaining eigenvalues are −1.

Definition: The phase gap of W (P) is ∆(P) = 2θ1.Fact: Relation between phase gap and eigenvalue gap of P:∆ ≥ 2

√δ.

From random to quantum walks [Szegedy’04]

Random walk

! P = (pxy )

! E-v: !k = cos "k! Stationary dist. (cos "0 = 1):

# = (#x )

! E-v gap: $ = 1! | cos "1|

1!1!

"1"2"3

#1

#2#3

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Spectral characterization of W (P) on the walk subspace

Theorem [Szegedy’04]: P ergodic, reversible Markov chain withstationary distribution π. Let the eigenvalues of P be:

1 = cos θ0 > cos θ1 ≥ . . . ≥ cos θk > −1Then, on the walk space, the spectrum of W (P) is:

• |π〉 =∑

x∈X

√πx |x〉|px〉 is the unique 1-eigenvector,

• e±2iθj are eigenvalues for 1 ≤ j ≤ k ,

• All remaining eigenvalues are −1.

Definition: The phase gap of W (P) is ∆(P) = 2θ1.Fact: Relation between phase gap and eigenvalue gap of P:∆ ≥ 2

√δ.

From random to quantum walks [Szegedy’04]

Random walk

! P = (pxy )

! E-v: !k = cos "k! Stationary dist. (cos "0 = 1):

# = (#x )

! E-v gap: $ = 1! | cos "1|

Quantum walk

! W = refY · refX! E-v (on X " Y): e±2i!k! Stationary state ("0 = 0):

|## =P

x

!"x |x"|px "

! phase gap: ! = |2"1|

0"

!

2!1

2!2

2!3

! = "($

$)%

quantum phase gap

= "!$classical e-v gap

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Quantum search

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Quantum search algorithms

Cost model of available procedures with data:

• Setup cost S: Construct the stationary state

|π〉 =∑

x∈X

√πx |x〉|px〉

• Update cost U: Realize the transformation and its inverse:

|x〉|0〉 7→ |x〉|px〉• Checking cost C: Reflection on marked elements:

refM : |x〉 7→{ |x〉 if x ∈ M−|x〉 otherwise

Update operations implement ref |x〉|px 〉, and therefore W (P) :

1 |x〉|px〉 7→ |x〉|0〉2 Reflect on |x〉|0〉3 Undo first step

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Grover search

Analogue of Search Algorithm 1

Grover search1 Prepare the stationary state: |π〉 =

∑x∈X

√πx |x〉|px〉

2 Repeat O(1/√ε)-times

1 Phase flip if first register is marked: −refM2 Reflect through |π〉 : ref |π〉

Theorem [Grover’96]: Final state is close to projection to markedelements |M〉 = 1√

ε

∑x∈M

√πx |x〉|px〉,

sinϕ =√ε

Grover’s algorithm

! We start with |!! = 1"|X |

!x!X |x!

! Goal: prepare |M! = 1"|M|

!x!M |x!

! We use 2 reflections:! through |M!!: refM! = #refM (C)! through |!!: ref! (S)

Grover’s algorithm

! Prepare |!! (S)

! Repeat T1$! apply refM! (C)! apply ref! (S)

Cost: 1"!(S+ C)

"

"

2" |##

|M#

|M!#

sin" = $M|##

=

s|M||X |

="

!

Cost: (S + C)/√ε

Idea: If S is high, replace ref |π〉 by some W (P)

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Ambainis searchAnalogue of Search Algorithm 2

Ambainis search1 Prepare |π〉2 Repeat 1/

√ε – times

1 Phase flip if first register is marked :−refM2 Perform 1/

√δ – steps of W (P)

3 Output first register

Theorem [Ambainis’04]: If P is the random walk on the Johnsongraph then it finds a marked element with high probability, at cost

S + 1√ε( 1√

δU + C).

Definition: The Johnson graph J(n, r) = (V ,E ):V = {S ⊆ {1, 2, . . . , n} : |S | = r}E : {S ,T} is an edge if they differ exactly in two elementsEigenvalue gap : δ = Θ(1/r)

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Szegedy search

Analogue of Search Algorithm 3

Szegedy search1 Prepare |π〉2 Repeat 1/(

√εδ) – times

1 Phase flip if first register is marked :−refM2 Perform one step of W (P)

3 Measure deviation from |π〉

Theorem [Szegedy’04]: If P is an ergodic, symmetric chain, then itdetects the presence of marked elements with high probability, atcost of

S + 1√εδ

(U + C).

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Magniez-Nayak-Roland-Santha searchIdea: Use W = W (P) to simulate ref |π〉 in Grover search

Back to Grover’s algorithmUses 2 reflections:

! Through marked states: refM! Through initial state: ref!

What if ref! is expensive?

=! Replace ref! by quantum walk W !

Reflexion ref!

0!

Quantum walkW

0!

!

2"1

2"22"3

Back to Grover’s algorithmUses 2 reflections:

! Through marked states: refM! Through initial state: ref!

What if ref! is expensive?

=! Replace ref! by quantum walk W !

Reflexion ref!

0!

Quantum walkW

0!

!

2"1

2"22"3

Let |π〉, |ψ1〉, . . . , |ψt〉 be the eigenvectors of W (P) in the walksubspace, and ∆ the phase gap. Spectral Theorem =⇒• ref |π〉|π〉 = |π〉 W |π〉 = |π〉• ref |π〉|ψj〉 = −|ψj〉 W |ψj〉 = e2iθj |ψj〉 where |2θj | ≥ ∆

Since ∆ ≥ 2√δ, we need a procedures which computes the phase

of the eigenvalues of W (P) with precision√δ

Phase estimation theorem [Kitaev’96, CEMM’98]: One can estimatewith k/

√δ applications of W (P) the phase of an eigenstate with

precision√δ and with error 2−k .

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MNRS search

MNRS search1 Prepare |π〉2 Repeat 1/

√ε steps of Grover search:

1 Phase flip if first register is marked :−refM2 Perform ref |π〉 with error

√ε:

1 Phase estimation for W (P) with precision√

δ and error√

ε2 If approximate phase is ∆-far from 0 then flip sign3 Undo phase estimation

3 Output first register

Theorem [Magniez-Nayak-Roland-Santha’07]: If P is an ergodic andreversible Markov chain, then it finds a marked element with highprobability, at cost of

S + 1√ε( log(1/ε)√

δU + C).

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Removing the log factor

How to deal with errors in Grover search?

• Errors in checking [Høyer-Mosca-de Wolf 03]Recursive version of Grover search, alternating• Amplitude amplification• Checking whose error decreases geometrically

Conclusion: Overall complexity is the same as without error

• Errors in diffusionSimilar recursion, using duality betweenChecking ↔ Diffusion, refM ↔ refπ• Amplitude amplification• Phase estimation based diffusion whose error decreases

geometrically

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MNRS Search

Theorem: If P is an ergodic and reversible chain, then the MNRSsearch finds a marked element with high probability, at cost of

S + 1√ε( 1√

δU + C).

Comparison with the other search algorithms:

• Conceptually simple and easy to analyze

• Unifies and generalizes previous approaches:• Ambainis search: Only for Johnson graph• Szegedy search: Only symmetric walk, only detects, run-time is

worse

• Improves several applications by log factors

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Applications in the quantum querymodel of computation

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Application 1

Element DistinctnessOracle Input: A function f defined on [n].Output: A pair of distinct elements i , j ∈ [n] such that f (i) = f (j)

Classically: nTheorem [Ambainis’04]: ED is solvable with n2/3 queries

P = Symmetric walk in the Johnson graph J(n, r)

x2x5 x7

x3

=

Element Distinctness via walk [Ambainis’04] 4

x2

x8 x13

x5 x2

x10 x13

x3

x2

x8 x13

x3

x2

x5 x13

x3

setup update checking

Quantum analogue- Speeds up both T1 and T2 quadratically [Ambainis‘04]

- Query complexity: r + (n/r) ( !r ! 2 + 0) ! n2/3 (optimal)

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Element Distinctness

• Marked element: R ⊆ [n] if there exist i 6= j ∈ R such that

f (i) = f (j)

• Parameters: ε = (r/n)2, δ = 1/r

• Data structure: d(R) = {(v , f (v)) : v ∈ R}• Costs: S = r ,U = 1,C = 0.

• Complexity:

S +1√ε

(1√δ

U + C) = r + n/r1/2 = n2/3

when r = n2/3.

Lower bound: [Aaronson-Shi’04]: n2/3

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Application 2Restricted Range AssociativityOracle Input: Operation ◦ : [n]× [n]→ [k] where k ∈ O(1).Output: Triple (a, b, c) such that (a ◦ b) ◦ c 6= a ◦ (b ◦ c)

Classically: n2

Theorem [Dorn-Thierauf’07]: RRA is solvable with n5/4 queries


• Marked element: R ⊆ [n] if there exist a, b ∈ R and c ∈ [n]such that (a ◦ b) ◦ c 6= a ◦ (b ◦ c)

• Parameters: ε = (r/n)2, δ = 1/r• Data structure: d(R) = {(a, b, a ◦ b) : a, b ∈ R ∪ [k]}• Costs: S = (r + k)2 = r2,U = r + k = r ,C = k

√rn =

√rn

• Complexity:

O(r2 + nr (√

r r +√

rn)) = n5/4

when r =√

n.

Lower bound (even with unrestricted range): n25/30

Application 3TriangleOracle Input: The adjacency matrix G of a graph on vertex set [n].Output: An edge of a triangle

Classically: n2

Theorem [Magniez-S-Szegedy’05]: T is solvable with n13/10 queries


• Marked element: R ⊆ [n] if it contains a triangle• Parameters: ε = (r/n)2, δ = 1/r• Data structure: d(R) = G |R• Costs: S = r2,U = r• Claim: Checking cost for Triangle is

√n × r2/3

• Complexity:

r2 + nr (√

r × r +√

n × r2/3) = n13/10

when r = n3/5.

Lower bound: n26/30

Triangle: Checking costClaim Checking cost for Triangle is

√n × r2/3

v

G|R

14Checking cost for Triangle edges

G|AG ! [n]2

G|A

Reduction: c(r) ! "n # QQC(Subproblem)

0

0

1

11

v0

v0 ! [n]

• G restricted to R, set of r vertices, is explicitly known• R is marked if an edge in R and some v form a triangle• Search over all v• For a fixed v , define secondary search problem

Input: Boolean fv on R where fv (u) = 1 if {u, v} is an edgeOutput: Edge {u, u′} in R such that fv (u) = fv (u′) = 1

• Analogous to Element Distinctness• Search in Johnson graph J(r , r2/3)• Complexity: r2/3

• Complexity of Checking is√

n × r2/3

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Application 4Group CommutativityOracle Input: Operation ◦ for a group whose base set contains [n].Output: A couple (i , j) ∈ [n]× [n] such that i ◦ j 6= j ◦ iClassically [Pak’00]: n4/3

Theorem [Magniez-Nayak’05]: GC is solvable with n2/3 log n queries

The walk P: Space is S(n, r)× S(n, r) where

S(n, r) = {u ∈ [n]r : ui 6= uj if i 6= j}.For i ∈ [r ], x ∈ [n] and u = (u1, . . . , ui , . . . , uj , . . . , ur ) ∈ S(n, r) let

ui,x =

{(u1, . . . , uj , . . . , ui , . . . , ur ) if x = uj

(u1, . . . , x , . . . , uj , . . . , ur ) otherwise

u

1/21/2nr

u11

urn

.

.

.

1/2nr

One step of P: Independently in both coordinates28/30

Group Commutativity

• For u = u1, . . . , ur , set u = (· · · (u1 ◦ u2) · · · ) ◦ ur

• Marked element: (u, v) if u ◦ v 6= v ◦ u.

• Parameters: ε = (r/n)2, δ = 1/(r log r)

• Data structure: d(u, v) = (Tu,Tv )

u1

r = 4

Database associated to a state 11

u

u1 ! u2 u3 ! u4

u3 u4u2

• Costs: S = r ,U = log r ,C = 1

• Complexity:

r + nr (√

r log r × log r + 1) = n2/3 log n

when r = n2/3 log n.

Lower bound: n2/3

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Conclusion

• Several quantum search algorithms

• MNRS search is conducive to composition of algorithms

• Extension: To ergodic, irreversible chains:eigenvalue 7→ singular value

• Phase estimation is used for NAND trees in [ACRSZ’07]

• Detection and search algorithms related to quantum hittingtime [MNRS08]

Open problems

• Exact complexity of• Matrix Product Verification• Associativity• Triangle

• Find other applications

• Schoning’s algorithm for 3-SAT

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quantum walk based search algorithms · quantum walks [meyer’96, watrous’98, aakv’01...

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