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 $η$-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 $\left(\frac{n^{5/3}}{η^{2/3}}+n^{4/3}η^{2/3}\right)n^{o(1)}$ gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $η$ electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when $n=η^{2-o(1)}$. 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.
Simulating Hamiltonian dynamics in a programmable photonic quantum processor using linear combinations of unitary operations
State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University（北京大学物理学院介观物理国家重点实验室）；Frontiers Science Center for Nano-optoelectronics & Collaborative Innovation Center of Quantum Matter, Peking University（北京大学纳米光子学与光电子前沿科学中心及量子物质协同创新中心）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）；Peking University Yangtze Delta Institute of Optoelectronics（北京大学长三角光电科学研究院）；Hefei National Laboratory（合肥国家实验室）
Beyond Trotterization: Variational Product Formulas for Quantum Simulation
Memorial University of Newfoundland and Labrador（纽芬兰与拉布拉多纪念大学）；Interdisciplinary Research Center (IRC) for Advanced Quantum Computing (AQC)（先进量子计算跨学科研究中心）；National Research Council Canada（加拿大国家研究委员会）；Perimeter Institute for Theoretical Physics（理论物理 perimeter 研究所）；Institute for Quantum Computing（量子计算研究所）；Compute Everything Technologies Ltd.（Compute Everything 科技有限公司）
Nearly Optimal T-Count for Symmetry-Based Quantum State Purification
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所处理器重点实验室）；School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）；Shandong University（山东大学）；Thrust of Artificial Intelligence, Information Hub, The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州）信息枢纽人工智能领域）
