  We introduce two quantum algorithms for solving structured prediction
problems. We first show that a stochastic gradient descent 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 then
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 variant of stochastic gradient descent 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.
