Ordering Matters: Structure, Accuracy and Gate Cost in Second-Order Suzuki Product Formulas

TL;DR

This paper investigates second-order Suzuki product formulas for quantum simulation, introducing new ordering strategies and fractional methods that improve accuracy and gate efficiency, especially under noise conditions.

Contribution

It introduces hybrid and fractional ordering methods for second-order Suzuki formulas, enhancing simulation accuracy and reducing gate costs in quantum computing.

Findings

Hybrid method achieves lowest error with high gate cost

Fractional decompositions often match or surpass hybrid performance with fewer gates

Fractional methods become advantageous under depolarising noise

Abstract

Product formula methods, particularly the second-order Suzuki decomposition, are an important tool for simulating quantum dynamics on quantum computers due to their simplicity and unitarity preservation. While higher-order schemes have been extensively studied, the landscape of second-order decompositions remains poorly understood in practice. We explore how term ordering and recursive application of the Suzuki formula generate a broad family of approximants beyond standard Strang splitting, introducing a hybrid heuristic that minimizes local error bounds and a fractional approach with tunable sequence length. The hybrid method consistently selects the longest possible decomposition, achieving the lowest error but at the cost of exponential gate overhead, while fractional decompositions often match or exceed this performance with far fewer gates, enabling offline selection of near-optimal approximants for practical quantum simulation. This offers a simple, compiler-accessible heuristic for balancing accuracy and cost, and highlights an underexplored region of decomposition space where many low-cost approximants may achieve high accuracy without global optimization. Finally, we show that in the presence of depolarising noise, fractional decompositions become advantageous as systems approach fault-tolerant error rates, providing a practical path for balancing noise resistance and simulation accuracy.