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arXiv:2007.05308

Fourth-order leapfrog algorithms for numerical time evolution of classical and quantum systems

Jun Hao Hue, Ege Eren, Shao-Hen Chiew, Jonathan Wei Zhong Lau, Chang, Leo, Thanh Tri Chau, Martin-Isbjörn Trappe, Berthold‐Georg Englert

Published Jul 10, 2020• 3 citations
arXivHTMLPDF

Abstract

Chau et al. [New J. Phys. 20, 073003 (2018)] presented a new and straight-forward derivation of a fourth-order approximation '$U_7$' of the time-evolution operator and hinted at its potential value as a symplectic integrator. $U_7$ is based on the Suzuki-Trotter split-operator method and leads to an algorithm for numerical time propagation that is superior to established methods. We benchmark the performance of $U_7$ and other algorithms, including a Runge-Kutta method and another recently developed Suzuki-Trotter-based scheme, that are exact up to fourth order in the evolution parameter, against various classical and quantum systems. We find $U_7$ to deliver any given target accuracy with the lowest computational cost, across all systems and algorithms tested here. This study is accompanied by open-source numerical software that we hope will prove valuable in the classroom.

DOI
10.48550/arxiv.2007.05308

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Hamiltonian formulations of lattice field theories provide access to real-time dynamics, but their simulation is difficult to implement efficiently. Trotter-Suzuki decompositions are at the center of time evolution computation, either on quantum hardware or classically, for instance with the use of tensor networks. While low-order Trotterizations remain the standard choice due to their simplicity, higher-order schemes offer the potential for improved efficiency. In this work we outline a short guide to Trotter-Suzuki schemes and their implementations in general. To help with this, we highlight new efficient schemes found by our optimization framework, and demonstrate their performance on the Heisenberg model.
The simulation of time evolution of large quantum systems is a classically challenging and often intractable task, making it a promising application for quantum computation. A Trotter-Suzuki approximation yields an implementation thereof, where a certain desired accuracy can be achieved by raising the gate count adequately. In this work, we introduce a variational algorithm which uses solutions of classical optimizations to predict efficient quantum circuits for time evolution of large quantum systems. Our strategy improves on the Trotter-Suzuki ansatz in accuracy and gate count by several orders of magnitude. In a NISQ-friendly setting, we can either significantly reduce the approximation error while using the same amount of gates as the Trotter-Suzuki ansatz or achieve a comparable accuracy with significantly fewer gates. We find, that our strategy outperforms the Trotter-Suzuki benchmark in translation invariant systems as well as for open boundaries.