Quantum Algorithms IIQuantum Algorithms II

Andrew C. Yao

Tsinghua University & Chinese U. of Hong Kong

Outline of Talk

I. Introduction

II. Element Distinctness & Sorting

III. Quantum Algorithm -- element distinctness

IV. Lower Bounds -- sorting problems

V. Conclusions

u

1e2e

Measurement

u1u

2u2

1 1

22 2

After the measurement

with probability | |

with probability | |

e u

e u

u

quantum state

Quantum MechanicsQuantum Mechanics

More generally

A quantum state is a unit vector u in CN

A measurement is a family of orthogonal

subspaces (V1, V2 , …,Vk):

u = u1+u2 +…+uk will be measured as ui with prob |ui|2

A computation step is a unitary operator (a ‘rotation’)

A Quantum Algorithm specifies

> an initial quantum state > a sequence of unitary operators > a final measurement.

Classical algorithm Quantum Algorithm

:

mapping

A M M ' :

Rotation

M MA C C

{0,1}mM

I.I. Introduction Introduction

Shor (‘94): Fast factorization of integers Grover (‘96): Searching n-item list

in time

Exactly one is 1, all other are 0

Problem: Find j

( )O n

1x 2x jx nx

jx ix

X

I.I. Introduction Introduction

Hilbert space with base vectors

-- Start with

-- Use a unitary operator to compute

(where ) Grover’s Theorem:

| , 1,2,...,i i n

01

1| > | >

n

i

u in

xU

0 1 2| > | > | > | >Tu u u u

x 1| > = | >i iu U u

|< | >| 1/10 for T ( ) =O Tj u n

I.I. Introduction Introduction

Initially,

After t steps,

Take measurement in base

the probability of seeing |j> is

0

1 1 1 1| > ( , ,... ,... )

un n

jn n

0 1 2| > | > | > | >Tu u u u

| > ( , ,... ,... )

t t t t

tu c c c

n

{|1>, | 2>, ..., | >}n

22

( )t t

nn

Random Walk

Hitting j by classical random walks takes n

steps

j

n

21

Quantum Random Walk

Hitting j by quantum random walks takes

steps

j

n

21

n

I.I. Introduction Introduction

Optimality Theorem (Bernstein et al ‘97): Any quantum algorithm must use steps to locate j

Question: What other problems can be speeded up with quantum algorithms?

This talk: Progress on sorting-like problems -- Element Distinctness Problem -- Sorting Problem

/100T n

I.I. Introduction Introduction Element Distinctness: Given decide whether there exist such that

Sorting: Given distinct determine such that

For conventional computers, essentially same problem (solvable in time ). For quantum computers, very different...

i j1, ... , nx x

i jx x

1 2...

ni i ix x x

1 , ... , nx x

logn n

1, ... , ni i

II.II. Element Distinctness & Sorting Element Distinctness & Sorting

Theorem 1. Element Distinctness can be solved in quantum steps.

* Buhrman et al ‘00 * Ambainis ‘03

Theorem 2. Any quantum algorithm for Element Distinctness must use time.

* Aaronson & Shi ‘04

2/3( )O n

2 / 3( )n

3/ 4( )O n2/3( )O n

II.II. Element Distinctness & Sorting Element Distinctness & Sorting

Sorting can be done in time classically.

Unlike Element Distinctness, it cannot be speeded up by quantum algorithms.

Theorem 3. Any quantum algorithm for Sorting must use time. * Hoyer, Neerbek and Shi ’02

Will illustrate Element Distinctness upper bound &

Sorting lower bound.

( log )n n

( log )O n n

III.III. Element Distinctness Element Distinctness

Quantum Algorithm for Element Distinctness

First, Grover’s list-searching implies: 0 0 1 1 0

Computing takes only evals.

Def: Element x in S is called a repeater if x = some x’ Question: Any repeater in S ?

3/ 4( )O n

( )O n

F1 F2 Fj Fn

1 2F F Fj Fn ( )O n

III.III. Element Distinctness Element DistinctnessQuantum Algorithm for Element Distinctness

1. Choose k, divide S into groups of size k

2. Design Repeateri : Output 1 iff some x in Si

is a repeater

3/ 4( )O n

1 1 2

2 1 2 2

/

: , , ... ,

: , ,... ,

. . . . . . . .

k

k k k

n k

S x x x

S x x x

S

III.III. Element Distinctness Element Distinctness

Repeater1: Any repeater in ?

1. Sort ; if there are equal elements, halt and return 1. 2. Use Grover to decide if some outside = some in ;

if yes, return 1

* Phase 1 takes time* Phase 2 takes quantum time to compute

ix

1S

1S

logk klogn k

1S

1 1 2 1 1( ) ( ) ( )k k nx S x S x S

x 1S

III. Element Distinctness III. Element Distinctness

Quantum algorithm for Element Distinctness: Use Grover to evaluate

1/ 2

3/ 4

Total time / time for one Search

( / ( log log ))

( ) log

(

log )

with the choice k

in k

O n k k k n k

nO k

O

n

n

n

n

kk

1 2 /Repeater Repeater Repeatern k

IV. Sorting for Partial Order PIV. Sorting for Partial Order P

P: partial order on n objects

Given input consistent with P Determine such that

The standard sorting problem is the special case P = empty

1 2, ,..., ni i i

1 2, ,..., nx x x

1 2 ni i ix x x

1 2( , ,..., )nx x x x

IV. Sorting Problem for Partial Order PIV. Sorting Problem for Partial Order P

e(P): # of linear extensions consistent with P Information bound:

Upper bound: [Fredman ‘75] [Kahn & Saks ‘91]

Quantum Complexity

Theorem 4 (Yao ‘04) constants

2log ( )e P

2log ( ) 2e P n2(log ( ))O e P

1 2 2log (( ) )cQ P e P c n

1 2, 0c c

( ) :Q P

Proof Outline

A. Quantum Partial Order Problem B. P=empty: Review of Proof

C. Reducing A to a Combinatorial Problem

D. Graph Entropy Chain Polytope

(lo

Sorting

g

Ta

(

kes

)) stepse P cn

A. Quantum Partial Order ProblemA. Quantum Partial Order Problem

= Linear extensions consistent with P Thus e(P) = | | Quantum algorithm S for partial order P computes the function

Note is a “partial function”

{0,1} ( )

1 if w ( 1), a

:

nd 0 otherwise

N

i ji

P

j

P

x xhere N n

f

n x

( )P( )P

Pf

Quantum Decision TreeQuantum Decision Tree

Initial state Unitary Operators Final measurement M

With prob > , M applied to gives the correct

1 2, ,..., TU U U

1 1 where

; , ( 1) ; ,i j

T x T x x

x

x

x U O U O U O

O z i j z i j

x1 ( )Pf x

For are ne , arl , y orthogonal x yx y

B. Review of CaseB. Review of Case

An entropy type argument (extending Ambainis)

At any time, the current state has an entropy (uncertainty). Initially, it has L= n log n bits of entropy. After the sorting, it should have 0 entropy. If each operation can reduce that entropy by at most , then the number of operations must be at least

P

constant

/ ( log )L n n

B. Review of CaseB. Review of Case [Hoyer, Neerbek, Shi ‘02][Hoyer, Neerbek, Shi ‘02]

For input x, after step j, the quantum state is

Initially all inputs x have states In the end, every pair x, y have nearly orthogonal states Define weight function w(x, y) for inputs x, y

Find lower bound to and upper bound to for any algorithm

1 1j x j x x

jx U O U O U O

P

,

1( , )

!j j

j x yx y

W w x yn

0 TW W

1j jW W

( )L

Then Q P

L

0x

, ' 0

1/ , ( ,( ) ),k d

and otherwisedw w xx xx

1

,

:

:

k k d n

k d

i i i i

cyclic shift

x x x x

B. Review of CaseB. Review of Case

Let

Lemma 1

Lemma 2

P permutation i t xnpu

0'TW W

1 2j jW W

( ) ( ) iff 1 iji j x

0 nW n H n

0(1 ')( ) ( log )

2

WQ n n

Extension toExtension to

Use the natural extension of their approach

Need to show a lower bound L to

P

', ' ( )

1( , ')

( )j j

j x xx x P

W w x xe P

0 TW W

0 log ( ) 'W c e P c n

C. Reduced to a Combinatorial ProblemC. Reduced to a Combinatorial Problem (*)

This is a purely combinatorial problem -- recall

(*) can be proved by using combinatorial

and information-theoretic tools developed

by Korner (‘73), Stanley (‘86), Kahn & Kim (‘95).

0 log ( ) 'W c e P c n

0, ' ( )

1( , ')

( ) x x P

W w x xe P

D. Graph Entropy, Chain Polytope, etc.D. Graph Entropy, Chain Polytope, etc. Proof outline for

The natural entropy for a partial order P would be simply log e(P).

Kahn and Kim (‘95) defined H(P), an alternative entropy (based on Korner’s graph entropy), and showed H(P) = (log e(P))

Our proof uses Stanley’s chain polytope theory to show that

0 log ( ) 'W c e P c n

0 ( )W H P

V. ConclusionsV. Conclusions

Quantum Complexity -- many open questions

Integer factorization has

classical polynomial time algorithm?

Graph isomorphism has

quantum polynomial time algorithm?

Rich source of problems: eg quantum complexity

for computing graph properties.