andris ambainis university of latvia...evaluating and-or trees variables x i accessed by queries to...
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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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Probabilistic computation Probabilistic system with
finite state space.
Current state: probabilities pi to be in state i.
1
2 3
4
0.6
0.1 0.2
0.1
i
ip 1
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Quantum computation Current state: amplitudes i
to be in state i.
1
2 3
4
0.4+0.3i
-0.7 0.4-0.1i
0.3
i
i 12
For most purposes, real (but negative) amplitudes suffice.
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Quantum computation Amplitude vector (1, …, M), .
Transitions:
before the transition
MMM
M
uu
uu
...
.........
...
1
111
transition matrix
M'
...
'1
after the transition
M
...
1
i
i 12
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Allowed transitions
U – unitary:
If , then .
MMM
M
uu
uu
...
.........
...
1
111
M'
...
'1
M
...
1
i
i 12
i
i 1'2
Equivalent to UU=I.
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Measurement
Quantum state:
1 |1 + 2 |2 + … + M |M
|1|2
1 prob. |2|
2
2 |M|2
M …
Measurement
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Quantum computing vs. nature Quantum computing Quantum physics
Unitary transformations U.
Transformation U performed in one step.
No intermediate states.
Physical evolution – continuous time.
Forces acting on a physical system – Hamiltonian H.
iHteU
Evolution for time t:
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Quantum algorithms up to 2005
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Shor’s algorithm
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.
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Grover's search
Find i such that xi=1.
Queries: ask i, get xi.
Classically, N queries required.
Quantum: O(N) queries [Grover, 96].
Speeds up any search problem.
0 1 0 0 ...
x1 x2 xN x3
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NP-complete problems Does this graph have a
Hamiltonian cycle?
Hamiltonian cycles are:
Easy to verify;
Hard to find (too many possibilities).
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Quantum algorithm
Let N – number of possible Hamiltonian cycles.
Black box = algorithm that verifies if the ith candidate – Hamiltonian cycle.
Quantum algorithm with O(N) steps.
0 1 0 0 ...
x1 x2 xN x3
Applicable to any search problem
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Pell’s equation Given d, find the smallest solution (x, y) to x2-dy2=1.
Probably harder than factoring and discrete logarithm.
Best classical algorithms:
for factoring;
2O(N) for Pell’s equation.
)( 3/1
2 NO
Hallgren, 2001: Quantum algorithm for Pell’s equation.
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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
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Formula evaluation
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AND-OR tree
AND
OR OR
x1 x2 x3 x4
OR OR
x5 x6 x7 x8
AND
OR
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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(
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Motivation Vertices = chess positions;
Leaves = final positions;
xi=1 if the 1st player wins;
At internal vertices, AND/OR evaluates whether the player who makes the move can win.
OR
x1 x2
How well can we play chess if we only know the position tree?
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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
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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.
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Augmented tree
0 1 1 0
Finite “tail” in one direction
…
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Finite tail algorithm
…
a -a -a a
Starting state:
j
j
start ja 2)1(
Hamiltonian H,
H – adjacency matrix
0 0
0 0
... ...
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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.
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When is the state unchanged? H – forces acting on the system.
(State | unchanged) H|=0.
e-iHt |Ψ = |Ψ H |Ψ = 0.
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What does H |Ψ = 0 mean?
…
H – adjacency matrix
a1 a2 a3
edgeji
ji
i
ab
bH
),(
,
H| = 0 for each i: 0),(
edgeji
ja
i
i ia
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T = 0 example
… a -a -a a 0 0
0
-a -a 0
0
0
0
0
OR
AND AND
1
0
1
0
| remains unchanged by H.
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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
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Summary [Farhi, Gutman, Goldstone, 2007] Hamiltonian
algorithm;
[A, Childs, et al., 2007] Discrete time algorithm.
O(N) time for full binary tree;
O(Nd) for any formula of depth d;
O(N1/2+o(1)) for any formula.
Improved to O(N log N) by [Reichardt, 2010].
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Span programs [Karchmer, Wigderson, 1993]
Target vector v.
Input x1, ..., xN vectors v1, ..., vM.
Output F(x1, ..., xN) = 1 if there ėxist vi1,vi2, ..., vik:
v=vi1+vi2+...+vik.
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Span program example
1
1
1
1
x1=1 x2=1 x3=1
1
0
Target
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Span program example
1
1
1
1
x1=1 x2=1 x3=1
1
0
Target
Output = 1.
x1=1, x2=1, x3=0
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Span program example
1
1
1
1
x1=1 x2=1 x3=1
1
0
Target
Output = 0.
x1=1, x2=0, x3=0
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Span program example
1
1
1
1
x1=1 x2=1 x3=1
1
0
Target
Output = “yes” if ≥2 of xi=1.
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Composing span programs Span program S1 with target t1.
Span program S2 with target t2.
Span program S1S2 with target t1+t2.
Answers 1 if both S1 and S2 answer 1.
F1, F2 F1 AND F2
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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
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Example MAJ(x1, x2, x3 , x4)=1 if at least 2 xi are equal to 1.
Formula size: 8.
Span program: 6.
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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
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Span programs [Reichardt, 2009]
Span program with witness size T
O(√T) query quantum algorithm
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Adversary bound [A, 2001, 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
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Span programs [Reichardt, 2009]
Optimal adversary bound
Semidefinite program (SDP)
Dual SDP
Optimal span program
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Span programs [Reichardt, 2009]
Span program with witness size T
O(√T) query quantum algorithm
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Summary Span programs = optimal quantum
algorithms.
Open problem: how to design good span programs?
Quantum algorithm for perfect matchings?
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Solving systems of linear equations
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The problem
Given aij and bi, find xi.
Best classical algorithm: O(N2.37...).
NNNNNN
NN
NN
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
...
2211
22222121
11212111
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Obstacles to quantum algorithm
Obstacle 1: takes time O(N2) to read all aij.
NNNNNN
NN
NN
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
...
2211
22222121
11212111
Solution: query access to aij.
Grover: search N items with O(N) quantum queries.
Obstacle 2: takes time O(N) to output all xi.
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Harrow, Hassidim, Lloyd, 2008
Output =
NNNNNN
NN
NN
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
...
2211
22222121
11212111
N
iiix
1
Measurement i with probability xi2.
Estimating c1x1+c2x2+...+cNxN.
Seems to be difficult classically.
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Harrow, Hassidim, Lloyd, 2008
Running time for producing : O(logc N), but with dependence on two other parameters.
Exponential speedup, if the other parameters are good.
N
iiix
1
NNNNNN
NN
NN
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
...
2211
22222121
11212111
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The main ideas
N
i
i ib1
N
i
i ix1
Easy-to-prepare Solution
NNNNNN
NN
NN
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
...
2211
22222121
11212111
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The main ideas
NNNN
N
N
aaa
aaa
aaa
A
...
............
...
...
21
22221
11211
Nx
x
x
x...
2
1
Nb
b
b
b...
2
1
bAx
bAx 1
N
i
i ib1
N
i
i ix1
How do we apply A-1?
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The main ideas
bAx 1
N
i
i ib1
N
i
i ix1
We can build a physical system with Hamiltonian A.
Unitary eiA.
eiA A-1 via eigenvalue estimation.
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Running time 1. Size of system N O(logc N).
2. Time to implement A – O(1) for sparse matrices A, O(N) generally.
3. Condition number of A.
min
max
k
max and min – biggest and smallest eigenvalues of A
NOTime clog2
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Dependence on condition number Classical algorithms for sparse A: O(Nk).
[Harrow, Hassidim, Llyod, 2008]: O(k2 logc N).
[A, 2010]: O(k1+o(1) logc N), via improved version of eigenvalue estimation.
[HHL, 2008]: (k1-o(1)), unless BQP=PSPACE.
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Open problem What problems can we reduce to systems of linear
equations (with as the answer)? i
i ix
Examples:
Search;
Perfect matchings in a graph;
Graph bipartiteness.
Biggest issue: condition number.