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.

翻译：估计哈密顿量的基态能量是一项基本任务，人们相信量子计算机对此任务有所帮助。为实现这一目标，已提出了多种方法，包括基于量子相位估计算法的方法，以及涉及参数化量子电路的混合量子-经典优化器方法，后者属于变分量子本征求解器的范畴。在此，我们分析了量子玻尔兹曼机在此任务中的性能，这是一种基于参数化热态的、较少被探索的拟设，且已知其不受贫瘠高原问题的影响。我们为此任务勾勒了一种混合量子-经典算法，并严格证明该算法收敛于能量函数在参数空间上优化的一个$\varepsilon$-近似驻点，同时使用的参数化热态样本数量在$\varepsilon^{-1}$、参数数量以及被优化的哈密顿量范数上是多项式级的。我们的算法通过一种新颖的量子电路构造高效地估计能量函数的梯度，该构造结合了经典采样、哈密顿量模拟和哈达玛测试，从而克服了自[Amin等人, Phys. Rev. X 8, 021050 (2018)]以来一直悬而未决的量子玻尔兹曼机器学习的一个关键障碍。进一步支持我们主要主张的，还包括对能量函数梯度和海森矩阵的计算，以及在收敛性分析中使用的对后者矩阵元素上界的推导。