Efficient Trotter-Suzuki Schemes Achieve Improved Long-Time Quantum Dynamics at Order 4

Scientists grapple with the persistent challenge of accurately modelling quantum systems over extended periods, hindered by the build-up of inaccuracies in standard computational methods. Marko Maležič and Johann Ostmeyer, from the Helmholtz-Institut für Strahlen- und Kernphysik, University of Bonn, alongside their colleagues, present a novel framework for designing highly efficient Trotter-Suzuki schemes to tackle this problem. Their research significantly advances the field by directly optimising these schemes, leading to the discovery of new decompositions that outperform traditional methods like those developed by Suzuki and Yoshida. This work not only recommends two particularly effective schemes , at fourth and sixth order , but also demonstrates their superior performance on benchmark models such as the Heisenberg model and harmonic oscillator, offering a pathway to more reliable and cost-effective long-time quantum dynamics simulations.

The team achieved the discovery of two new schemes, operating at 4th and 6th order, exhibiting significantly improved performance compared to existing methods. This breakthrough reveals a method for identifying the optimal structure of high-order Trotter-Suzuki schemes, moving beyond simply improving existing low-order approximations. Researchers employed a parameter optimisation strategy, allowing them to explore a vast design space and uncover schemes with enhanced efficiency, a crucial step towards tackling complex simulations.

This finding provides valuable insight into the characteristics of robust Trotter decompositions, guiding future development in the field. The team’s framework isn’t limited to quantum systems; it also extends to classical simulations, offering potential benefits for molecular dynamics and other areas of physics. Furthermore, the research opens avenues for improving simulations of complex physical systems, including lattice gauge theories and condensed matter models, which have previously been hampered by computational limitations. The ability to accurately simulate these systems is crucial for advancing our understanding of fundamental physics and developing new technologies. Though the study focuses on unitary time evolution, the framework could potentially be adapted for non-unitary methods, broadening its impact even further. Ultimately, this work promises to enhance the accuracy and efficiency of numerical simulations across a wide range of scientific disciplines, enabling researchers to explore previously inaccessible regimes of complex systems.

High-order Trotter schemes for optimised dynamics simulations

This work directly addresses the limitations of low-order decompositions, which, while easy to implement, quickly become inaccurate for long-time simulations due to rapidly escalating errors. The study harnessed advanced computational techniques to explore the high-dimensional parameter space, enabling the discovery of optimal parameter sets for the Trotter-Suzuki decompositions. This precise optimisation process delivers a substantial improvement in the accuracy and efficiency of long-time dynamics simulations, opening new avenues for exploring complex physical systems. Furthermore, the research extends beyond purely quantum systems, demonstrating the applicability of these optimised schemes to classical simulations involving Hamiltonian dynamics and phase-space distributions. The team highlights the connection to symplectic integrators, commonly used in molecular dynamics, noting a key difference in optimisation due to the vanishing nested commutator structure, which simplifies the process. This versatility underscores the broad impact of their methodological advancements, offering a powerful tool for researchers across diverse fields of physics and computational science.,.

Optimised Trotter schemes enhance long-time dynamics simulations

Based on theoretical efficiency and practical performance, the team recommends two new schemes at order four and order six, demonstrating substantial advancements in computational accuracy. The team measured the theoretical efficiency of these decompositions, defining it as Effn = 1/qnErrn, where Errn represents the leading errors of a scheme and qn is the number of cycles. Results demonstrate that the newly developed schemes exhibit a plateau towards maximal efficiency as the number of cycles increases, indicating a fundamental limit to improvement. At order four with six cycles, and order six with fourteen cycles, the recommended schemes closely align with the theoretically most efficient decompositions, although those decompositions sometimes performed worse in practice.

Tables 2 and 3 detail the parameters ci for these schemes, providing a resource for implementation and further study. Data shows that solutions are possible with fewer parameters than constraints starting from order six, creating “accidental” solutions that do not expand into one-dimensional manifolds even with increased cycles. Researchers found that for orders ten and above, only these accidental solutions exist, limiting the potential for free-parameter optimisation. The framework was benchmarked by successfully recovering historical schemes, then applied to unexplored cycle regions, revealing the theoretical efficiency of these decompositions.

The team consistently identified global minima in the error function Errn, confirming the robustness of the optimisation process. Specifically, the Heisenberg model was simulated using a Hamiltonian with couplings Jx = Jy = Jz = 1 and a magnetic field sampled from a uniform distribution between -0.1 and 0.1. Measurements confirm a slight theoretical.