Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit $\Lambda$ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most $\epsilon$ can be achieved by choosing $\Lambda$ to scale polylogarithmically with $\epsilon^{-1}$, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on $\Lambda$ that achieves accuracy $\epsilon$, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy $\epsilon$ by truncating local quantum numbers at $\Lambda=\textrm{polylog}(\epsilon^{-1})$.

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规范理论和玻色子系统的可证明准确的模拟

涉及玻色子模式或规范场的量子多体系统具有无限维的局部希尔伯特空间，必须将其截断以在经典或量子计算机上执行实时动力学模拟。为了分析截断误差，我们开发了限制局部量子数增长率的方法，例如晶格位置上模式的占有数，或晶格链接处的电场。我们的方法适用于玻色子与自旋或费米子相互作用的各种模型，也适用于阿贝尔和非阿贝尔规范理论。我们表明，如果通过对每个局部量子数施加上限 $\Lambda$ 来截断这些模型中的状态，并且如果初始状态具有较低的局部量子数，然后，通过选择 $\Lambda$ 与 $\epsilon^{-1}$ 进行多对数缩放，最多可以实现一个误差 $\epsilon$，这是对基于能量守恒的先前界限的指数改进。对于 Hubbard-Holstein 模型，我们在数值上计算了 $\Lambda$ 上的界限，该界限达到了准确度 $\epsilon$，在各种参数方案中获得了显着改进的估计。我们还建立了一个截断哈密顿量的标准，可证明时间演化的准确性。基于该结果，我们制定了用于动态模拟晶格规范理论和具有玻色子模​​式的模型的量子算法；在前一种情况下，门的复杂性几乎与时空体积呈线性关系，在后一种情况下几乎与时间呈二次方关系。我们建立了一个下限，表明存在涉及玻色子的系统，无法改进这种随时间的二次缩放。通过将我们的结果应用于时间演化中的截断误差，我们还证明了通过在 $\Lambda=\textrm{polylog}(\epsilon^{- 1})$。