# Smooth Structured Prediction Using Quantum and Classical Gibbs Samplers

By Behrooz Sepehry, Ehsan Iranmanesh, Michael P. Friedlander, & Pooya Ronagh
We introduce two quantum algorithms for solving structured prediction problems. We show that a stochastic subgradient descent method that uses the quantum minimum finding algorithm and takes its probabilistic failure into account solves the structured prediction problem with a runtime that scales with the square root of the size of the label space, and in $\widetilde {O}\left(1/\epsilon\right)$ with respect to the precision, $\epsilon$, of the solution. Motivated by robust inference techniques in machine learning, we introduce another quantum algorithm that solves a smooth approximation of the structured prediction problem with a similar quantum speedup in the size of the label space and a similar scaling in the precision parameter. In doing so, we analyze a stochastic gradient algorithm for convex optimization in the presence of an additive error in the calculation of the gradients, and show that its convergence rate does not deteriorate if the additive errors are of the order $O(\sqrt\epsilon)$. This algorithm uses quantum Gibbs sampling at temperature $\Omega (\epsilon)$ as a subroutine. Based on these theoretical observations, we propose a method for using quantum Gibbs samplers to combine feedforward neural networks with probabilistic graphical models for quantum machine learning. Our numerical results using Monte Carlo simulations on an image tagging task demonstrate the benefit of the approach.

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