Exact quantum Exact quantum algorithmsalgorithms

Andris AmbainisAndris Ambainis

University of LatviaUniversity of Latvia

Types of quantum algorithmsTypes of quantum algorithms

Bounded-error: correct answer with Bounded-error: correct answer with probability at least 2/3.probability at least 2/3.

Exact: correct answer with certainty Exact: correct answer with certainty (probability 1).(probability 1).

Grover's searchGrover's search

Is thereIs there i i::xxii=1=1?? Classically, N queries required.Classically, N queries required. Quantum: O(Quantum: O(N) queries [Grover, 96].N) queries [Grover, 96]. Quantum, exact: N queries.Quantum, exact: N queries.

0 1 0 0...

x1 x2 xNx3

ModelModel

Query modelQuery model

Function f(xFunction f(x11, ..., x, ..., xNN), x), xii{0,1}.{0,1}.

xxii given by a black box: given by a black box:

i xi

Complexity = number of queries

Queries in the quantum worldQueries in the quantum world

Basis sBasis statestates:: |1 |1,1,1, |, |1, 1, 22, …, |N, …, |N, M, M.. Query:Query:

||i, ji, j ||i, ji, j, if x, if xii=0;=0;

||i, ji, j -| -|i, ji, j, if x, if xii=1;=1;

ExampleExample

1,11,1|1|1, 1, 1++1,21,2||1, 1, 22++2,12,1||2, 12, 1++3,13,1||3,13,1

0 1 0

x1 x2 x3

Query

1,11,1|1|1, 1, 1++1,21,2||1, 1, 22- - 2,12,1||2, 12, 1++3,13,1||3,13,1

Quantum query modelQuantum query model

Fixed starting state.Fixed starting state. UU00, U, U11, …, U, …, UTT – independent of x – independent of x11, …, x, …, xNN..

Q – queries.Q – queries. Measuring final state gives the result.Measuring final state gives the result.

U0 Q QU1 UT…

Known exact algorithmsKnown exact algorithms

Deutsch’s problemDeutsch’s problem

Determine xDetermine x11xx22, with query access to x, with query access to x ii..

[Cleve et al., 1998]: 1 quantum query, [Cleve et al., 1998]: 1 quantum query, always the correct answer. always the correct answer.

0 1x1 x2

Dutsch-JozsaDutsch-Jozsa

Distinguish whether:Distinguish whether: xx11 = x = x22 = ... = x = ... = xNN or or

xxii=0 (x=0 (xii=1) for exactly ½ of i=1) for exactly ½ of i{1, 2, ..., N}. {1, 2, ..., N}.

Deterministic: N/2+1 queries.Deterministic: N/2+1 queries. Quantum: 1 query.Quantum: 1 query.

x1 x2 xNx3

0 1 0 0...

Grover's searchGrover's search

Is thereIs there i i::xxii=1=1?? Promise: there is 0 or 1 i: Promise: there is 0 or 1 i: xxii=1=1.. ClassicallyClassically:: N queries N queries.. QuantumQuantum, exact, exact: O(: O(N) queriesN) queries..

x1 x2 xNx3

0 1 0 0...

Exact algorithms for total Exact algorithms for total functions?functions?

Deutsch’s problemDeutsch’s problem

Determine xDetermine x11xx22, with query access to x, with query access to x ii..

[Cleve et al., 1998]: 1 quantum query, [Cleve et al., 1998]: 1 quantum query, always the correct answer. always the correct answer.

0 1x1 x2

x1x2...xN can be computed with N/2 queries

Montanaro et al., 2011.Montanaro et al., 2011.

EXACTEXACT2244(x(x11, x, x22, x, x33, x, x44)=1 if there are )=1 if there are

exactly 2 i:xexactly 2 i:xii=1.=1.

Classical: 4 queries.Classical: 4 queries. Quantum: 2 queries, exact.Quantum: 2 queries, exact.

Is there a total function f(x1, ..., xN) for which QE(f) < D(f)/2?

quantum exact deterministic

Our resultsOur results

Superlinear separationSuperlinear separation

TheoremTheorem There is f(x There is f(x11, ..., x, ..., xNN) such that) such that D(f)=N;D(f)=N; QQEE(f)=O(N(f)=O(N0.86...0.86...).).

What should f be?

Polynomial degree lower bound Polynomial degree lower bound

deg(f) – degree of f(xdeg(f) – degree of f(x11, ..., x, ..., xNN) as a ) as a

multilinear polynomial. multilinear polynomial. [Nisan, Szegedy, 92, Beals et al., 98][Nisan, Szegedy, 92, Beals et al., 98]

Basis functionBasis function

D(f)=3, deg(f)=2

Iterated NEIterated NE

x11x22 x33

NE

NE NENE

x44x55 x66 x77

x88 x99

d levels D(f)=3d, deg(f)=2d

Our resultOur result

TheoremTheorem For d levels, Q For d levels, QEE(f)=O(2.593...(f)=O(2.593...dd).).

x11x22 x33

NE

NE NENE

x44x55 x66 x77

x88 x99

Step 1Algorithm for NE(x1, x2, x3). Starting state:

Result:

Step 2Step 2

p-algorithm:p-algorithm: ||startstart | |startstart if f=0; if f=0;

||startstart p| p|startstart + | + | with | with |||startstart, if f=1., if f=1.

p=0 exact quantum algorithm

Step 3Step 3 p-algorithm:p-algorithm:

||startstart | |startstart if f=0; if f=0;

||startstart p| p|startstart + | + | with | with |||startstart, if f=1., if f=1.

NE(x1, x2, x3) – 2 queries, p = -7/9

f

p-algo, k queries

f

NE

f f

p’-algo, 2k queries

Step 3: resultStep 3: result

x11x22 x33

NE

NE NENE

x44x55 x66 x77

x88 x99

d levels, 3d variables;

p-algorithm with 2d queries.Bad p!

Step 4Step 4

Amplification

f

p-algo, k queries 2k queries, smaller p

f

Form of amplitude amplification [Brassard et al., 2000]

Final algorithmFinal algorithm

1 level, 3 variables, 2 queries

Iterate

2 levels, 9 variables, 4 queries

Iterate

3 levels, 27 variables, 8 queries

Amplify

3 levels, 27 variables, 16 queries...

Final resultFinal result

221111 queries for each 8 levels. queries for each 8 levels. N=3N=388 variables, 2 variables, 21111 queries. queries. N=3N=38k8k variables, 2 variables, 211k11k queries. queries.

QE(f)=N0.86...

Other exact quantum Other exact quantum algorithmsalgorithms

EXACTEXACT

Determine whether xDetermine whether xii=1 for exactly k of N =1 for exactly k of N variables.variables.

Montanaro et al., 2011:Montanaro et al., 2011: Algorithm: 2 out of 4, 2 queries;Algorithm: 2 out of 4, 2 queries; Computer optimization: 3 out of 6, 3 queries;Computer optimization: 3 out of 6, 3 queries; Conjecture: N/2 out of N, N/2 queries. Conjecture: N/2 out of N, N/2 queries.

0 1 0 0...

x1 x2 xNx3

A, Iraids, SmotrovsA, Iraids, Smotrovs

Exact algorithms for determining:Exact algorithms for determining: if xif xii=1 for exactly N/2 i, N/2 queries;=1 for exactly N/2 i, N/2 queries;

if xif xii=1 for exactly k i, max(k, N-k) queries;=1 for exactly k i, max(k, N-k) queries;

Provably optimal.Provably optimal.

Natural computational problems; Simple algorithms.

Algorithm: summaryAlgorithm: summary

1 query

1 query

... ...

Threshold functionsThreshold functions

Is it true that xIs it true that xii=1 for =1 for k of N variables?k of N variables?

Exact algorithm, max(k, N-k+1) queries.Exact algorithm, max(k, N-k+1) queries. Easiest: kEasiest: k==N/2, N/2+1 queries.N/2, N/2+1 queries. Hardest: k=0 or k=N, N queries.Hardest: k=0 or k=N, N queries.

0 1 0 0...

x1 x2 xNx3

SummarySummary

A function that requires N queries A function that requires N queries classically, O(Nclassically, O(N0.86...0.86...) queries for exact ) queries for exact quantum algorithms. quantum algorithms.

First separation by more than a factor of 2.First separation by more than a factor of 2. Several other exact quantum algorithms.Several other exact quantum algorithms.

Advantages for exact quantum algorithms are more common that I thought

Open problems Open problems

1.1. d-level NE function (with 3d-level NE function (with 3dd variables): variables): O(2.593...O(2.593...dd) query exact algorithm;) query exact algorithm; Lower bound: Lower bound: (2.11...(2.11...dd).).

2.2. Other iterated functions?Other iterated functions?

3.3. Other symmetric functions?Other symmetric functions?

4.4. More exact algorithms?More exact algorithms?

Open problemsOpen problems

5.5. Lower bound methods for exact quantum Lower bound methods for exact quantum algorithms?algorithms?

Currently known:Currently known: Bounded-error quantum lower bounds;Bounded-error quantum lower bounds; QQEE(f) (f) deg(f)/2; deg(f)/2;

For NEFor NEdd, both of them fail., both of them fail.

More informationMore information

A. Ambainis. Superlinear advantage for A. Ambainis. Superlinear advantage for exact quantum algorithms, exact quantum algorithms, arxiv:1211.0721.arxiv:1211.0721.

A. Ambainis, J. Iraids, J. Smotrovs. A. Ambainis, J. Iraids, J. Smotrovs. Exact quantum query complexity of EXACT and THRESHOLD, arxiv:1302.1235.rxiv:1302.1235.