New methods for quantum algorithms...New methods for quantum algorithms Andris Ambainis University of Latvia Datorzinātnes lietojumi un tās saiknes ar kvantu fiziku AQIS’2011,

New methods for quantum

algorithms

Andris Ambainis

University of Latvia

Datorzinātnes lietojumi un tās saiknes ar kvantu fiziku

AQIS’2011, August 25, 2011

Shor’s algorithm [1994]

Factoring: given N=pq, find p and q.

Best algorithm - 2O(n1/3), n – number of digits.

Quantum algorithm - O(n3) [Shor, 94].

Cryptosystems based on hardness of

factoring/discrete log become insecure.

Grover's search [1996]

Find i such that xi=1.

Queries: ask i, get xi.

Classically, N queries required.

Quantum: O(N) queries [Grover, 96].

0 1 0 0 ...

x1 x2 xn x3 ...

Speeds up any search problem.

Element distinctness [A, 2004]

Numbers x1, x2, ..., xN.

Determine if two of them are equal.

Classically: N queries.

Quantum: O(N2/3).

7 9 2 1 ...

x1 x2 xN x3

Evaluating logic formulas [Farhi, et

al., 2007, Reichardt, 2011]

AND

OR

AND x1 x2

OR

x3 x4

AND

x5 x6

Formula of size N.

O(√N) quantum

algorithms.

From algorithms to methods

Element distinctness:

Search by a quantum walk (quantization of Markov chains);

Formula evaluation:

Span programs;

Mathematical structures

Quantum query model

Input data x1, ..., xN given by a black box

that answers queries;

Q |i |i if xi=0

-|i if ja xi=1

Complexity = number of queries.

Examples: Grover, element distinctness, etc.

Quantum algorithm

Start

state

Q U1 Q ...

End

state

Part 1

Search by quantum walk

Generic search problem

Finite search space.

Some states marked.

Task: find a marked

state.

1

2 3

4 Solution: random walk

on search space.

Search by a random walk

Start in random state;

Perform a random

walk, until finding a

marked state.

T – expected number

of steps.

1

2 3

4

Szegedy, 2004: Quantum walk detects

a marked state in O(√T) steps.

In more detail...

Random walk:

T walking steps to find a marked state;

Time complexity: S+TW:

S – time to generate a random starting state;

W – time to perform one walking step;

Szegedy, 2004: Quantum walk,

O(S+√TW ) steps.

Matrix multiplication [Buhrman,

Špalek, 05]

A, B, C – n*n matrices.

Finding C=AB: O(n2.37…) steps;

Given A, B and C, we can test AB=C in:

O(n2) steps by a probabilistic algorithm;

O(n5/3) steps by a quantum algorithm.

Triangle finding [Magniez, Santha,

Szegedy, 05]

Graph G with n vertices.

n2 variables xij; xij=1 if there

is an edge (i, j).

Does G contain a triangle?

Classically: O(n2).

Quantum: O(n1.3).

Forbidden subgraph properties

[Childs, Kothari, 2011]

P=“G contains one of several

subgraphs H1, ..., Hk”.

P – sparse if any G not

containing H1, ..., Hk has O(n)

edges

Any sparse P can be decided

in o(n3/2) steps.

Search vs. finding

What can we do in time O(S+√TW )?

[Szegedy, 2004] Quantum algorithm for

detecting whether a marked state exists.

*Assumes knowledge of p, ppmarked,

pmarked – fraction of marked states.

[Krovi et. al., 2010] Quantum algorithm for

finding a marked state*.

Part 2

Quantum algorithms for formula

evaluation

Evaluating AND-OR trees

Variables xi accessed by queries to a black box:

Input i;

Black box outputs xi.

Quantum case:

Evaluate T with the smallest number of queries.

OR

AND AND

x1 x2 x3 x4

i

x

i

i

i iaia i)1(

Results (up to 2007)

Full binary tree of depth d.

N=2d leaves.

Deterministic: (N).

Randomized [SW,S]:

(N0.753…).

Quantum?

Easy q. lower bound:

(N).

OR

AND AND

x1 x2 x3 x4

New results

[Farhi, Gutman, Goldstone, 2007]:O(N) time

algorithm for evaluating full binary trees in

Hamiltonian query model.

[A, Childs, Reichardt, Spalek, Zhang, 2007]:

O(N1/2+o(1)) time algorithm for evaluating any

formulas in the usual query model.

[Reichardt, 2011] O(√N) query algorithm.

0 1 1 0

Finite “tail” in one direction

…

Augmented tree

…

a -a -a a

Starting state:

Hamiltonian H,

H – adjacency matrix

0 0

0 0

... ...

Farhi et al. algorithm

What happens?

If T=0, the state stays almost unchanged.

If T=1, the state “scatters” into the tree.

Run for O(N) time, check if the state |Ψ is close to the starting state |Ψstart.

When is the state unchanged?

H – forces acting on the system.

(State | unchanged) H|=0.

…

H – adjacency matrix

a1 a2 a3

H| = 0 for each i: 0),(

edgeji

ja

What does H|=0 mean?

Example

OR

AND AND

1

0

1

0

Formula Augmented tree

H| = 0 state

… a -a -a a 0 0

0

-a -a 0

0

0

0

0

OR

AND AND

1

0

1

0

Such state can be constructed whenever T=0.

General property

… a -a -a a 0 0

0

-a -a 0

0

0

0

0

OR

AND AND

1

0

1

0

Leaves with non-zero ai force T=0.

T=1 case

… a -a -a a 0 0

0

-a -a 0

0

0

0

0

AND

OR OR

1 1 1

0

No | with H|=0.

0

Cannot place non-zero value here

Span programs [Reichardt, Špalek, 2008]

Span program with witness size T

O(√T) query quantum algorithm

Logic formula of size T

Far-reaching generalization of formula evaluation

Span programs [Karchmer, Wigderson, 1993]

Target vector v.

Input x1, ..., xN vectors v1, ..., vM.

Output F(x1, ..., xN) = 1 if there exist

vi1,vi2, ..., vik:

v=vi1+vi2+...+vik.

Span programs [Reichardt, 2011]

Span program with witness size T

O(√T) query quantum algorithm

Adversary bound [A, 2000, Hoyer, Lee, Špalek,

2007]

Boolean function f(x1, ..., xN);

Inputs x = (x1, ..., xN);

Matrix A: A[x, y]≠0 only if f(x) ≠ f(y)

Theorem Computing f requires

quantum queries

Span programs [Reichardt, 2009]

Optimal adversary bound

Semidefinite program (SDP)

Dual SDP

Optimal span program

Span programs [Reichardt, 2011]

Span program with witness size T

O(√T) query quantum algorithm

What can we do with

span programs?

Example

MAJ(x1, x2, x3 , x4)=1 if at least 2 xi are equal

to 1.

Formula size: 8.

Span program: 6.

Iterated thresholds

MAJ

x1 x2 x3 x4

MAJ

x5 x6 x7 x8

MAJ MAJ

... ...

MAJ

d levels – formula of size 8d, span program 6d.

O(6d) quantum algorithm

Singularity testing

Matrix A;

Promise A is singular or all singular values of A

are at least min.

Task: distinguish between the two cases.

Singularity testing

1, 2, ..., N - singular values of A.

Theorem [Belovs, 2011] Quantum algorithm for

testing singularity in time where

av

NTO

~

N

i

iavN 1

2

min

2,max

1

T – time to implement A as a Hamiltonian

Triangle finding

Graph G with n vertices.

Does G contain a triangle?

[Belovs, 2011]: O(n1.29...).