We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of the Hamiltonian terms within the $\eta$-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use $\left(\frac{n^{5/3}}{\eta^{2/3}}+n^{4/3}\eta^{2/3}\right)n^{o(1)}$ gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $\eta$ electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when $n=\eta^{2-o(1)}$. We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.
翻译:我们考虑在数字量子计算机上模拟量子系统。 我们显示量子模拟的性能可以通过同时利用目标汉密尔顿式的通气性、互动的广度和初始状态的先前知识来改进。 我们用Trotter化技术实现, 包括各种物理系统, 包括平浪基电子结构和Fermi- Hubbbard模型。 我们用美元旋转轨道和美元元电子元来利用蜂巢式汉密尔顿语客运客的过渡振幅来估计模拟模拟错误。 我们开发了多种技术, 用来约束一般 fermionic操作员的过渡振幅和预期, 这可能具有独立的兴趣。 我们显示, 足够使用美元(frac{n ⁇ 5/3 ⁇ %2 ⁇ } {n__________________________________________ral__________________________________________________________________________/_____________________________________________________________________________________________________________________________________________________________________________________________________________________________