Quantum Boltzmann machine learning of ground-state energies
This addresses a fundamental task in quantum computing for researchers, offering a novel method with rigorous convergence guarantees, though it is incremental as it builds on prior work in quantum Boltzmann machines.
The paper tackles the problem of estimating ground-state energies of Hamiltonians by analyzing quantum Boltzmann machines, a less explored ansatz that avoids the barren-plateau issue, and proves that their hybrid quantum-classical algorithm converges to an ε-approximate stationary point with polynomial sample complexity.
Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an $\varepsilon$-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in $\varepsilon^{-1}$, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a novel quantum circuit construction that combines classical sampling, Hamiltonian simulation, and the Hadamard test, thus overcoming a key obstacle to quantum Boltzmann machine learning that has been left open since [Amin et al., Phys. Rev. X 8, 021050 (2018)]. Additionally supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.