Quantum lower

bounds and group

representation theory

Andris Ambainis

University of Latvia

European Social Fund project “Datorzinātnes pielietojumi un tās saiknes ar kvantu fiziku” Nr.2009/0216/1DP/1.1.1.2.0/09/APIA/VIAA/044

Query model

Input x1, …, xN accessed by queries.

Complexity = the number of queries.

0 1 0 0 ...

x1 x2 xN x3

i

0

i

xi

i

x

i

i

i iaia i1

Grover's search

Is there i such that xi=1?

Queries: ask i, get xi.

Classically, N queries required.

Quantum: O(N) queries [Grover, 1996].

0 1 0 0 ...

x1 x2 xN x3

Quantum speed-up for any search problem.

Element distinctness

Are there i, j such that ij but xi=xj?

Classically: N queries.

Quantum: O(N2/3) [A, 2004].

3 1 17 5 ...

x1 x2 xN x3

Triangle finding

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).

[Belovs, 2011] Quantum:

O(n1.29...).

Lower bounds

Search requires N) queries [Bennett et

al., 1997].

Element distinctness: (N2/3) [Shi, 2002].

Triangle finding: (N) [easy].

Lower bound methods

Adversary: analyze algorithm, prove it is

incorrect on some input.

Polynomials: describe algorithm by low degree

polynomial.

History of adversary method

[Bennett, et al., 1997] Hybrid argument, (√N)

lower bound for quantum search.

[A, 2000] Adversary method, first general lower

bound theorem.

[Barnum, Saks, Szegedy, 2003] Spectral

adversary method.

[A, 2003, Zhang, 2004] Weighted adversary.

[Laplante, Magniez, 2004] Kolmogorov

complexity.

History of adversary method (2)

[Špalek, Szegedy, 2005] spectral, weighted and

Kolmogorov complexity methods are all

equivalent.

[Hoyer, Lee, Špalek, 2007] weighted adversary

with negative weights.

[Reichardt, 2009, 2011] weighted adversary with

negative weights is optimal.

Reichardt, 2011

F(x1, ..., xN) – computational problem.

T - best quantum lower bound for F provable by

negative-weight adversary method.

Theorem There is a quantum algorithm A that

computes f with O(T) queries.

Proof ideas (1)

Method for quantum algorithms:

Span programs [Reichardt, Špalek, 2008];

Method for quantum lower bounds:

Negative-weight adversary [Hoyer, Špalek, Lee,

2007];

Maximizing the parameters in both methods =

semidefinite program (generalization of linear

program).

Proof ideas (2)

Semidefinite programming duality:

Min (Primal program) = Max (Dual program).

Primal program = Span program size;

Dual program = Adversary lower bound.

Implies optimality for both methods.

Does this solve every problem?

No, we still have to find:

the best span program;

the best adversary lower bound parameters.

Index erasure

Distinct x1, x2, …, xN{1, 2, …, M}, access by queries.

Generate the state

Motivation: graph isomorphism.

3 1 17 5 ...

x1 x2 xN x3

N

i

ixN 1

1

Index erasure

Easy to generate

N

i

ixiN 1

1

N

i

iN 1

1

Erasing |i takes O(N) queries.

No better solution known.

Quantum lower bound?

Index erasure

Quantum algorithm: O(√N) queries.

[Midrijanis, 2004]: (N1/5/logcN) lower bound

for set equality (which reduces to index erasure).

[A, Magnin, Roettler, Roland, 2011, this talk]

(√N) lower bound for index erasure.

Previous adversary

method

Quantum query model

Fixed starting state.

U0, U1, …, UT – independent of x1, x2, …, xN.

Q – queries.

Measuring final state gives the result.

U0 Q Q start U1 UT …

Queries

Basis states for algorithm’s workspace: |i, z,

i{1, 2, …, N}.

Query transformation:

Example:

|i, z|i, z, if xi=0;

|i, z-|i, z, if xi=1;

zQiziQix,

|

Adversary framework

Quantum algorithm A

x1 x2 … xN

NxxN xxxQxxxN

...... 21...21 1

Two registers: HA, HI.

Query Q:

Example:Grover search

Start state: |start|0,

End state

1...00...0...010...101

0 N

1...00...0...0120...1011

NN

Density matrices

Measure HA, look at density matrix of HI

N

N

N

end

100

01

0

001

NNN

NNN

NNN

start

111

111

111

New method

State of algorithm’s knowledge

State:

| Quantum

algorithm A x1 x2 … xN

...21 2211 IAIA

State |1 quantifies algorithm’s knowledge

about the input if A is in state |1.

State of algorithm’s knowledge

1...00...0...010...101

0 N

| Quantum

algorithm A x1 x2 … xN

A has no information about the location of xi=1.

1...00...0...0120...1011

NN

A has perfect information about the location of xi=1.

Symmetries of the problem

Let - permutation of {1, 2, …, N}.

Run algorithm on x(1), x(2), …, x(N):

Query to xi replaced by query to x(i).

0 1 0 0 ...

x1 x2 xN x3

Symmetries of the problem

For any algorithm A, there is another

algorithm A’:

A’ has the same success probability as A.

State in |x1 x2 … xN register of A’ symmetric.

| Quantum

algorithm A x1 x2 … xN

Example: Grover search

Grover search; inputs

|10…0, |01…0, …,

|00…1.

t - state of HI after t

steps.

abb

bab

bba

t

State of any search algorithm can be described by two parameters: a and b.

Group representations

Representation theory

Group G, linear space H.

For each element gG, linear transformation

Ug: H H.

Transformations satisfy Ugh= Ug Uh.

representation of G

Irreducible representation: no decomposition

H=H1H2, Ug:H1H1, Ug:H2H2.

|

Proof

Adversary framework

H – linear space consisting of all

Quantum algorithm A

x1 x2 … xN

N

N

xxx

Nxxx xxx

21

21 21

Symmetries

G – group of symmetries of f(x1, ..., xN).

Representation of G, for example:

Ug:|x1 x2 … xN |x(1)x(2) … x(N).

Use representation theory to decompose

H = H1 H2 ... Hk,

Hi –irreducible representations.

If t – state of |x1 x2 … xN after t steps,

t – invariant under all Ug.

Strategy

H = H1 H2 ... Hk,

Hi –irreducible and invariant.

i – completely mixed state over Hi.

Claim If t – state of |x1 x2 … xN after t steps,

then

t = pt,1 1 + pt,2 2 +... + pt,k k.

complete description of the algorithm

Examples

Example 1: Grover’s search

States of the input register

1|10...0+2|01...0+... +n|00...1

H=H0H1.

State after t steps: t = p 0 + (1-p) 1.

H0 – no information about i:xi=1.

H1 – full information about i:xi=1.

Example 2: k-fold search [A, 2006]

k locations i:xi=1.

Task: find all of them.

States of the input register

H=H0H1... Hk.

Hj – algorithm knows j of k locations i:xi=1.

State after t steps: t = p00 + ... + pkk.

Kxi

Nxxx

i

Nxxx

|}1:{|

2121

Index erasure

Distinct x1, x2, …, xN{1, 2, …, M}, access by queries.

Task: generate the state

3 1 17 5 ...

x1 x2 xN x3

N

i

ixN 1

1

Symmetries for index erasure

Input states |x1, x2, …, xN.

Two types of symmetries.

Permuting indices of x1, x2, …, xN.

Permuting values 1, 2, …, M.

Symmetry group SNSM.

What are the irreducible representations?

Young diagrams

N squares;

In each row, the number of

squares is at most the number

for the previous row.

Young diagrams with N squares

representations of SN

Representations of SNSM

Pairs of Young diagrams (one for SN, one for

SM).

For the index erasure problem, the diagram for

SN must be contained in the diagram for SM:

Informal interpretation

Corresponds to the algorithm knowing:

4 of values xi;

Locations i for 3 of those 4 values.

3 4

Examples

321321

1,,,,

:,...,

21 ,...,,

yyyxxxxx

n

iii

n

xxx

States of the form

yxxx

nyi

i

n

xxx:,...,

21,

1

,...,,

1.

2.

Adversary argument

Start with |0|start,

If algorithm succeeds, the final state is

},...,1{,...,

1

1

,...,Mxx

Nstart

N

xx

Nxx

N

i

ifinal xxxN,...,

1

1

,...,1

Irreducible representations

Starting state:

Final state: N M

Main result

Let t – state of A after t queries. Then, the

probability of representations

is at least

N

tO1

same shape

Hence, (√N) queries are required.

Conclusion

New quantum lower bound method, based on symmetries and analysis of group representations.

An optimal (√N) lower bound for index erasure problem.

Several related problems remain open.

Open problem 1: 3-distinctness

Are there i1, i2, i3 such that xi1= xi2

= xi3?

Classically: N queries.

3 1 17 5 ...

x1 x2 xN x3

Quantum: O(N5/7) [Belovs, 2012].

Quantum lower bound: (N2/3) [from

element distinctness).

Open problem 2: set equality

Promise:

x1, ..., xN are all different;

y1, ..., yN are all different;

{x1, ..., xN} and {y1, ..., yN} are either equal or

different;

Task: are they equal or different?

3 1 5 ...

x1 x2 xN ...

6 7 4 ...

y1 y2 yN ...

Open problem 2: set equality

Task: are {x1, ..., xN} and {y1, ..., yN} equal or

different?

Quantum algorithm: O(N1/3) [collision];

Q. lower bound: (N1/5/logcN) [Midrijanis, 2004]

Related to maximum speedup for symmetric

functions [Aaronson, A, 2011].

3 1 5 ...

x1 x2 xN ...

6 7 4 ...

y1 y2 yN ...

Open problem 3: graph properties

Graph G on n vertices;

Variables xij, xij=1 if there is an edge (i, j).

Function f(G), does not depend on the order of

vertices;

E.g., f(G) = 1 if G contains a triangle.

What is the smallest possible complexity of a

monotone graph property?

Bounds: O(N), (N2/3).