Recent experiments demonstrated quantum computational advantage in random circuit sampling and Gaussian boson sampling. However, it is unclear whether these experiments can lead to practical applications even after considerable research effort. On the other hand, simulating the quantum coherent dynamics of interacting spins has been considered as a potential first useful application of quantum computers, providing a possible quantum advantage. Despite evidence that simulating the dynamics of hundreds of interacting spins is challenging for classical computers, concrete proof is yet to emerge. We address this problem by proving that sampling from the output distribution generated by a wide class of quantum spin Hamiltonians is a hard problem for classical computers. Our proof is based on the Taylor series of the output probability, which contains the permanent of a matrix as a coefficient when bipartite spin interactions are considered. We devise a classical algorithm that extracts the coefficient using an oracle estimating the output probability. Since calculating the permanent is #P-hard, such an oracle does not exist unless the polynomial hierarchy collapses. With an anticoncentration conjecture, the hardness of the sampling task is also proven. Based on our proof, we estimate that an instance involving about 200 spins will be challenging for classical devices but feasible for intermediate-scale quantum computers with fault-tolerant qubits.

翻译：近期实验证明了随机电路采样和高斯玻色采样中的量子计算优势。然而，这些实验能否在大量研究投入后产生实际应用尚不明确。另一方面，模拟相互作用自旋的量子相干动力学一直被视为量子计算机首个潜在实用应用，可能提供量子优势。尽管有证据表明，模拟数百个相互作用自旋的动力学对经典计算机具有挑战性，但具体证明仍有待给出。我们通过证明一类广泛量子自旋哈密顿量产生的输出分布采样对经典计算机而言是困难问题，解决了这一难题。我们的证明基于输出概率的泰勒级数，其中当考虑二分自旋相互作用时，级数系数包含矩阵的积和式。我们设计了一种经典算法，利用估计输出概率的预言机提取该系数。由于计算积和式属于#P-难问题，除非多项式层级坍缩，否则此类预言机不存在。基于反集中猜想，采样任务的困难性同样得到证明。根据我们的证明，我们估计涉及约200个自旋的实例将对经典设备具有挑战性，但对具有容错量子比特的中等规模量子计算机而言是可行的。