Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. As a result, when Nature cools a quantum system in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, Nature finds a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. While local minima are much easier to find than ground states, we show that finding a local minimum is computationally hard for classical computers, even when the task is to output a single-qubit observable at any local minimum. In contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics the cooling process in Nature. To establish the classical hardness of finding local minima, we consider a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding ground states of these Hamiltonians. We prove that for such Hamiltonians, all local minima are global minima. Therefore, assuming quantum computation is more powerful than classical computation, finding local minima is classically hard and quantumly easy.

翻译：寻找量子多体系统的基态被认为对经典计算机和量子计算机而言都是困难的。因此，当自然系统在低温热浴中冷却时，基态并非总能被有效找到。取而代之的是，自然系统找到能量的局部极小值。在本工作中，我们研究了在热扰动下量子系统中寻找局部极小值的问题。尽管局部极小值比基态更容易找到，但我们证明，即使任务只是输出任意局部极小值下的单个量子比特可观测量，对于经典计算机而言，寻找局部极小值在计算上仍然是困难的。相比之下，我们证明了量子计算机可以通过模拟自然冷却过程的热梯度下降算法始终高效地找到局部极小值。为了确立寻找局部极小值的经典困难性，我们考虑了一族二维哈密顿量，使得任何可由多项式时间量子算法解决的问题都可以归约为寻找这些哈密顿量的基态。我们证明，对于这类哈密顿量，所有局部极小值都是全局极小值。因此，假设量子计算比经典计算更强大，则寻找局部极小值在经典上困难而在量子上容易。