The computational complexity of simulating the dynamics of physical quantum systems is a central question at the interface of quantum physics and computer science. In this work, we address this question for the simulation of exponentially large bosonic Hamiltonians with quadratic interactions. We present two results: First, we introduce a broad class of quadratic bosonic problems for which we prove that they are $\mathsf{BQP}$-complete. Importantly, this class includes two known $\mathsf{BQP}$-complete problems as special cases: Classical oscillator networks and continuous-time quantum walks. Second, we show that extending the aforementioned class to even more general quadratic Hamiltonians results in a $\mathsf{PostBQP}$-hard problem. This reveals a sharp transition in the complexity of simulating large quantum systems on a quantum computer, as well as in the difference in complexity between their simulation on classical and quantum computers.

翻译：模拟物理量子系统动力学的计算复杂度是量子物理与计算机科学交叉领域的核心问题。本文针对具有二次相互作用的指数大规模玻色子哈密顿量的模拟问题展开研究，取得两项成果：首先，我们定义了一类广义二次玻色子问题，并证明其具有$\mathsf{BQP}$完全性。值得注意的是，该问题类包含两个已知的$\mathsf{BQP}$完全问题作为特例：经典振子网络与连续时间量子游走。其次，我们证明将上述问题类扩展至更一般的二次哈密顿量将导致$\mathsf{PostBQP}$困难问题。这一发现揭示了量子计算机模拟大规模量子系统复杂度的陡峭跃迁，以及经典计算机与量子计算机在模拟此类系统时存在的本质复杂度差异。