超越高斯动力学的玻色系统李代数经典模拟
Lie-Algebraic Classical Simulation of Bosonic Systems Beyond Gaussian Dynamics
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中文总结 AI 辅助
该研究突破李代数模拟的常规局限，建立了玻色系统的精确多项式时间经典模拟框架，可高效计算相关可观测量，覆盖高斯量子光学并拓展至非高斯动力学，为玻色量子动力学的研究提供了统一形式体系。
中文摘要 AI 辅助
经典可模拟性最终由量子系统的动力学和所评估的可观测量共同决定。李代数模拟利用后者，通过在低维不变算子空间中传播可观测量，实现精确的多项式时间经典模拟。然而，其基于多项式维动力李代数的常规表述无法直接适配玻色系统，因为玻色系统的李代数既非紧致也非半单。在本研究中，我们克服了这一限制，使玻色系统可纳入李代数框架下的精确多项式时间经典模拟范畴。我们证明，当期望、固定阶关联函数（包括多时间关联函数和非时间序关联函数）以及梯度对应的算子模块具有多项式维时，这些量可高效计算。这一结果覆盖了高斯量子光学，并将其扩展至非高斯输入态，同时确定了含非高斯相互作用动力学的精确多项式 regime，包括有界光子克尔哈密顿量、对跃哈密顿量以及幂零多项式相位动力学。我们表明，与先前研究的有限维自旋和费米子情形不同，仅有限维玻色生成元代数无法保证有限可观测量动力学。我们进一步推导了超出精确 sector 约束的压缩受控微扰层级，并通过数值验证了预测的误差阶。此外，我们对多达 400 个模式的相互作用链上的算子扩散进行了评估，并将拓扑 doublon 带与通量反转的边缘运动关联起来。这些结果为利用经典多项式时间模拟对可处理的玻色量子动力学进行分类、发现及系统近似提供了统一形式体系。
英文摘要
Classical simulability is ultimately determined by both the dynamics of a quantum system and the observables being evaluated. Lie-algebraic simulation exploits the latter to make exact polynomial-time classical simulations by propagating observables through low-dimensional invariant operator spaces. However, its conventional formulation in terms of polynomial-dimensional dynamical Lie algebras does not directly accommodate bosonic systems as their algebras are neither compact nor semisimple. In this contribution, we overcome this limitation, making bosonic systems accessible to the Lie-algebraic programme of exact polynomial-time classical simulation. We prove that expectation values, fixed-order correlation functions, including multi-time correlators and out-of-time-ordered correlators, and gradients are efficiently computable whenever their operator modules have polynomial dimension. This recovers Gaussian quantum optics and extends it to non-Gaussian input states, while identifying exact polynomial regimes of interacting non-Gaussian dynamics including bounded-photon Kerr and pair-hopping Hamiltonians and nilpotent polynomial phase dynamics. We show that unlike in the finite-dimensional spin and fermionic setting treated previously, a finite-dimensional bosonic generator algebra alone does not guarantee finite observable dynamics. We further derive a controlled perturbative hierarchy for squeezing beyond exact sector confinement and confirm the predicted error orders numerically. We also evaluate operator spreading on interacting chains of up to $400$ modes and connect a topological doublon band with flux-reversed edge motion. These results provide a unified formalism for classifying, discovering, and systematically approximating tractable bosonic quantum dynamics with classical polynomial-time simulation.
