Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions

Abstract

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.

Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation

“…The only previously known algorithm for non-destructive amplitude estimation is given in [HW19]. It works via several invocations of amplitude estimation according to [BHMT00], which is based on phase estimation.…”

“…• Our new algorithm requires dramatically fewer ancillae. This is because [HW19] relies on phase estimation with median amplification. As argued above, median amplification requires O(n log(δ −1 )) ancillae.…”

“…• Our new algorithm runs in a fixed amount of time: one application of our energy estimation algorithm suffices. In contrast, [HW19]'s algorithm works by repeatedly attempting to 'repair' the input state. This process succeeds only with probability 1/2, so while the expected number of attempts is constant, the resulting algorithm is highly adaptive and may need a variable amount of time.…”

“…where |λ j is an n-bit estimate of λ j . This access to spectral information enables quantum speedups for linear algebra [HHL08,CKS15], studying physical systems [Temme&09, YA11, Lemi&19, JKKA20], estimating partition functions [Mon15], and performing Bayesian inference [HW19,AHNTW20]. Phase estimation is also a very complicated algorithm.…”

“…However, for some applications of phase estimation maintaining coherence remains crucial. The original strategy for quantum matrix inversion [HHL08], quantum Metropolis sampling [Temme&09,YA11,Lemi&19], and a protocol for partition function estimation [Mon15], thermal state preparation, and Bayesian inference [HW19,AHNTW20] all violate the assumptions above. The eigenvalues must be estimated while preserving the superposition, and there is no guarantee that the input state is an eigenstate.…”