Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule.
We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution.
In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.
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Quantum eigenstates from classical Gibbs distributions
We discuss how the language of wave functions (state vectors) andassociated noncommuting Hermitian operators naturally emerges fromclassical mechanics by applying the inverse WignerWeyl transform to thephase space probability distribution and observables. In this language,the Schr"odinger equation follows from the Liouville equation, with \hbar ℏ now a free parameter. Classical stationary distributions can berepresented as sums over stationary states with discrete (quantized)energies, where these states directly correspond to quantum eigenstates.Interestingly, it is now classical mechanics which allows for apparentnegative probabilities to occupy eigenstates, dual to the negativeprobabilities in Wigner’s quasiprobability distribution. These negativeprobabilities are shown to disappear when allowing sufficientuncertainty in the classical distributions. We show that thiscorrespondence is particularly pronounced for canonical Gibbs ensembles,where classical eigenstates satisfy an integral eigenvalue equation thatreduces to the Schr"odinger equation in a saddlepointapproximation controlled by the inverse temperature. We illustrate thiscorrespondence by showing that some paradigmatic examples such astunneling, band structures, Berry phases, Landau levels, levelstatistics and quantum eigenstates in chaotic potentials can bereproduced to a surprising precision from a classical Gibbs ensemble,without any reference to quantum mechanics and with all parameters(including \hbar ℏ )on the order of unity.
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 Award ID(s):
 1813499
 NSFPAR ID:
 10293278
 Date Published:
 Journal Name:
 SciPost Physics
 Volume:
 10
 Issue:
 1
 ISSN:
 25424653
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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