Quantum lower bounds and group representation theory · 2012. 6. 18. · Element distinctness Are...
Transcript of Quantum lower bounds and group representation theory · 2012. 6. 18. · Element distinctness Are...
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).