quantum lower bounds and group representation theory · 2012. 6. 18. · element distinctness are...
TRANSCRIPT
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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
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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
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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.
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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
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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...).
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Lower bounds
Search requires N) queries [Bennett et
al., 1997].
Element distinctness: (N2/3) [Shi, 2002].
Triangle finding: (N) [easy].
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Lower bound methods
Adversary: analyze algorithm, prove it is
incorrect on some input.
Polynomials: describe algorithm by low degree
polynomial.
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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.
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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.
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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.
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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).
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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.
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Does this solve every problem?
No, we still have to find:
the best span program;
the best adversary lower bound parameters.
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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
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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?
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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.
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Previous adversary
method
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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 …
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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,
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|
Adversary framework
Quantum algorithm A
x1 x2 … xN
NxxN xxxQxxxN
...... 21...21 1
Two registers: HA, HI.
Query Q:
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Example:Grover search
Start state: |start|0,
End state
1...00...0...010...101
0 N
1...00...0...0120...1011
NN
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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
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New method
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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.
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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.
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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
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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
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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.
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Group representations
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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.
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|
Proof
Adversary framework
H – linear space consisting of all
Quantum algorithm A
x1 x2 … xN
N
N
xxx
Nxxx xxx
21
21 21
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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.
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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
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Examples
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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.
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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
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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
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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?
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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
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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:
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Informal interpretation
Corresponds to the algorithm knowing:
4 of values xi;
Locations i for 3 of those 4 values.
3 4
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Examples
321321
1,,,,
:,...,
21 ,...,,
yyyxxxxx
n
iii
n
xxx
States of the form
yxxx
nyi
i
n
xxx:,...,
21,
1
,...,,
1.
2.
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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
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Irreducible representations
Starting state:
Final state: N M
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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.
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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.
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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).
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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 ...
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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 ...
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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).