Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. By organizing error contributions according to spatial support rather than Taylor order, we compute local finite-time contributions numerically while bounding only the nonlocal tails analytically. For translation-invariant one-dimensional chains, our bounds overestimate exact second-, fourth-, and sixth-order Trotter errors by roughly 40–50% on average in our tests, improving on existing higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.

Pizza and drinks will be served after the seminar in ATL 2117.