Quantum advantage in variational Bayes inference

Abstract

Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm—inspired by variational methods used in computational physics—is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a VB inference algorithm based on a nontraditional quantum annealing approach—referred to as quantum annealing variational Bayes (QAVB) inference—and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key quantum mechanics concepts: i) The ground state of the Hamiltonian of a quantum system—defined from the given data—corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using ⌈log K ⌉ qubits and 𝒪( K ) operations per step, where K is the number of values hidden categorical variables can take. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.