Quantum Simulated Annealing

21 Pages·2008·0.71 MB·English

Quantum Simulated Annealing
Howard Barnum1 (presenting)
&
RolandoSomma2 SergioBoixo1,3 MannyKnill4
1LosAlamosNationalLaboratory
2PerimeterInstitute
3UniversityofNewMexico
4NationalInstituteofStandardsandTechnology(Boulder)
March 28, 2008 / Classical and Quantum Information
Theory, Santa Fe, NM
Introduction and outline
Motivation
Markov chain Monte-Carlo (MCMC) algorithms, such as
simulated annealing (SA) are important in statistical physics
and optimization. Can we get a quantum speed-up?
Barnum QuantumSimulatedAnnealing
Combinatorial optimization
Problem instance
Minimize a nonnegative objective function E, over a search
space S.
S is a set of d elements, called configurations.
S may have additional structure, e.g. providing a notion of
locality.
d typically is exponential in the size of the problem
instance.
Notation:
S : the set of minimal-energy configurations.
0
E : E −E ; typically polynomial in instance size
M max min
γ: the classical gap between E and the next-lowest
min
energy.
E /γ is a natural parameter: dynamic range / resolution.
M
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Combinatorial Optimization II: Examples
S the set of paths on a graph with n vertices whose edges
are assigned lengths, E(σ) the length of path σ.
S the set of spin configurations (assignments
σ = (σ ,σ ,....,σ ) of σ = ±1 to each vertex) on an N-site
1 2 N i
lattice L whose edges are colored with nearest-neighbor
interaction energies h , d = 2N, E the Ising Hamiltonian
ij
(cid:88) (cid:88)
E(σ) = h σσ + Bσ . (1)
ij i j i i
<i,j>∈L i
Typically NP-hard.
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Combinatorial optimization (III)
With d exponential in problem size, exhaustive search is bad
news.
When objective function reflects structure—e.g. notion of
locality or distance—in S, other methods may outperform
exhaustive search.
E.g., with something like global convexity, gradient-like methods
can help.
Can we do better when some “smoothness”, but no “global
convexity”?
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Simulated annealing
Simulated annealing starts system at high “temperature”, and
cools to “ground space” S . “Thermal excitations” at
0
intermediate temperatures intended to pop system out of local
minima.
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Simulated Annealing (II)
Start hot: cool and sample from the resulting Boltzmann-Gibbs
distribution concentrated on S .
0
Choose an annealing
schedule β < β < ... < β .
1 2 P
Choose {M(β )}. Each M(β )
k k
is a stochastic matrix with
Boltzmann-Gibbs equilibrium
distribution(cid:126)π,
(cid:126)π = M(β )(cid:126)π , πσ = e−βE[σ]/Z .
k
At each step
σ(k−1) M−(→βk) σ(k) .
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Example of transition rule: Metropolis
Start with σ and repeat:
0
1 Select test state σ(cid:48) with rule g
2 Jump to σ(cid:48) with rule Aσ(cid:48)σ
(cid:26) e−β∆E ∆E > 0
A =
σ(cid:48)σ 1 otherwise
Figure:MarkovchainMonte-CarlowithMetropolisrule
Average after mixing Mixing time τ (Aldous)
mix
For δ the gap of M(β) ≡ Ag.
(cid:80)
O(σ )
(cid:104)O(cid:105) = f
#runs δ−1 ≤ τmix ≤ δ−1(ln1/π∗)
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SA convergence and cost
(cid:32) (cid:33)
P
(cid:89)
Final state µ(cid:126)(β ) = M(β ) µ(cid:126)(0) .
f k
k=1
Convergence: Cool slowly enough (rate proportional to
gap).
(cid:88)d (µσ(β ))2
∆β = O(δ/E ) ⇒ f ≤ 2 .
M πσ(β )
f
σ=1
Error from the temperature: Cool enough.
(cid:88)
πσ(β ) ≤ de−βfγ ≤ (cid:15)2 .
f
σ∈/S
0
SA cost
(cid:18)E log(d/(cid:15)2)(cid:19)
O(β /∆β) = O(β E /δ) = O M
f f M γ δ
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Adiabatic Computation
Farhietal.,quant-ph/0001106
Use the adiabatic theorem to go from an “easy” ground state to
the solution ground state.
(cid:16) s(cid:17) s
H(s) = 1− H + H .
Easy Problem
T T
Adiabatic condition
Move slower than the slowest
frequencies (gap squared).
(cid:107)H˙(cid:107)/δ2 ≤ (cid:15) ⇒ T = O(1/((cid:15)δ2)) .
AQC ≡ BQP
Adiabatic quantum computation is general quantum
computation.
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