A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

doi:10.1007/s42967-023-00336-z

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (2): 442-469.doi: 10.1007/s42967-023-00336-z

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A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

Shi Jin¹, Xiantao Li²

1 School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China;
2 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA

收稿日期:2023-04-21 修回日期:2023-09-29 接受日期:2023-10-03 出版日期:2025-06-20 发布日期:2025-04-21
通讯作者: Xiantao Li,Xiantao.Li@psu.edu;Shi Jin,shijin-m@sjtu.edu.cn E-mail:Xiantao.Li@psu.edu;shijin-m@sjtu.edu.cn
基金资助:
Jin’s research was supported by the NSFC (Grant No. 12031013) and the Innovation Program of Shanghai Municipal Education Commission (Nos. 2021-01-07-00-02-E00087). Li’s research has been supported by the NSF (Grant Nos. DMS-1953120 and DMS-2111221).

A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

Shi Jin¹, Xiantao Li²

1 School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China;
2 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA

Received:2023-04-21 Revised:2023-09-29 Accepted:2023-10-03 Online:2025-06-20 Published:2025-04-21
Supported by:
Jin’s research was supported by the NSFC (Grant No. 12031013) and the Innovation Program of Shanghai Municipal Education Commission (Nos. 2021-01-07-00-02-E00087). Li’s research has been supported by the NSF (Grant Nos. DMS-1953120 and DMS-2111221).

摘要/Abstract

摘要： Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising, respectively, from the random sampling of the Hamiltonian terms and the operator-splitting error. Leveraging on the bias/variance tradeoff, the error can be minimized by balancing the two. The concentration bound provides an estimate of the number of gates. The estimates are verified using numerical experiments on classical computers.

关键词: Trotter splitting, Quantum simulation, Unitary dynamics

Abstract: Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising, respectively, from the random sampling of the Hamiltonian terms and the operator-splitting error. Leveraging on the bias/variance tradeoff, the error can be minimized by balancing the two. The concentration bound provides an estimate of the number of gates. The estimates are verified using numerical experiments on classical computers.

Key words: Trotter splitting, Quantum simulation, Unitary dynamics

中图分类号:

Shi Jin, Xiantao Li. A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 442-469.

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A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

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