Constant-Depth Multi-Product Formula for Trotter Error Mitigation in Near-Term Digital Quantum Simulation

Digital quantum simulations of many-body dynamics are limited by a trade-off between algorithmic Trotter error (which decreases with circuit depth) and physical noise (which increases with circuit depth). The authors investigate whether higher-order product formulas (HOPFs) provide a genuine advantage on noisy hardware and propose a new mitigation strategy that avoids the depth-inflation typical of traditional approaches.

The authors benchmark various higher-order product formulas against a second-order Leapfrog baseline, using two-qubit circuit depth as a proxy for the physical noise budget. They then develop the constant-depth multi-product formula (cd-MPF). This approach uses an auxiliary parameter to reshape Trotter error terms while keeping the target evolution invariant. By taking a classical linear combination of expectation values from circuits at the same fixed depth, the method cancels the leading-order Trotter error term.

Standard higher-order formulas often fail to outperform the simpler Leapfrog formula on noisy hardware because their increased gate requirements amplify physical noise. In contrast, the cd-MPF successfully suppresses the leading-order Trotter error while maintaining the same physical noise profile as the baseline. Benchmarks across multiple spin models (TFIM, XXZ, XY) demonstrate that cd-MPF consistently achieves lower observable errors than both Leapfrog and higher-order formulas at matched circuit depths, provided the simulation is within the perturbative regime where the leading-order error dominates.

This work provides a practical path for extending the reach of near-term quantum simulators. By decoupling algorithmic error suppression from circuit depth, the cd-MPF allows researchers to achieve higher accuracy in long-time dynamics simulations without incurring the prohibitive physical noise penalties associated with deeper circuits.