exact quantum algorithms andris ambainis university of latvia
TRANSCRIPT
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.