Trotter error time scaling separation via commutant decomposition

Suppressing the Trotter error in dynamical quantum simulation typically requires running deeper

circuits, posing a great challenge for noisy near-term quantum devices. Studies have shown that

the empirical error is usually much smaller than the one suggested by existing bounds, implying the

actual circuit cost required is much less than the ones based on those bounds. Here, we improve the

estimate of the Trotter error over existing bounds, by introducing a general framework of commutant

decomposition that separates disjoint error components that have fundamentally different scaling

with time. In particular we identify two error components that each scale as O(t^p+1/r^p) and

O(t^p/r^p) for a pth-order product formula evolving to time t using r partitions. Under a fixed step

size t/r, it implies one would scale linearly with time t and the other would be constant of t. We

show that this formalism not only straightforwardly reproduces previous results but also provides

a better error estimate for higher-order product formulas. We demonstrate the improvement both

analytically and numerically. We also apply the analysis to observable error relating to the heating

in Floquet dynamics and thermalization, which is of independent interest.