Optimising Trotter-Suzuki Simulations of Markovian Open Quantum Systems via Classical Search

Simulating open quantum systems (OQS) on digital quantum computers is essential for understanding non-isolated quantum dynamics. A primary challenge is the computational cost associated with Trotter-Suzuki (TS) product formulas, which approximate the system's time evolution. This paper addresses the gap between overly conservative theoretical (analytic) bounds on the number of Trotter steps required for a target precision and the actual number of steps needed in practice.

The researchers derive explicit analytic bounds for four simulation methods: First- and Second-Order Deterministic and Randomized TS-PFs. To improve upon these, they introduce a classical algorithm that utilizes diamond norm estimates of individual Liouvillian terms combined with a binary search to identify the minimum number of Trotter steps required for a specific target precision. They validate this approach on two prototypical models: an XX-Spin Chain with boundary driving and local dephasing, and a Transverse-Field Ising Model (TFIM).

The study reveals that theoretical analytic bounds consistently overestimate the number of Trotter steps needed to achieve a target precision. By using the proposed empirical optimization algorithm, the researchers achieve a significantly smaller number of steps, leading to reduced gate complexity. Among the methods tested, the Second-Order Randomized TS-PF is identified as the most resource-efficient, particularly as the system size increases. The authors also demonstrate that the diamond norm of local Liouvillian terms can be computed efficiently, avoiding the exponential scaling typically associated with such calculations.

These findings provide a practical framework for optimizing quantum simulations. By moving from conservative theoretical estimates to empirical, model-specific bounding strategies, researchers can perform more complex simulations on current and near-term quantum hardware with fewer gates, thereby reducing the impact of hardware noise and limited coherence times.