Trotter Error Analysis
Where product-formula error comes from, how it depends on commutators, and how to bound it rigorously.
The origin of the error
For H = A + B, the exact evolution and the first-order product differ because the Baker-Campbell-Hausdorff expansion mixes A and B. Expanding both to second order, e^(-i(A+B)t) - e^(-iAt) e^(-iBt) = -(t^2/2)[A,B] + O(t^3). The leading error is proportional to the commutator [A,B] = AB - BA.
Why commutators, not norms
A naive bound replaces the commutator by 2||A|| ||B||, which is often far too pessimistic. If A and B nearly commute, the real error is tiny even when their norms are large. This is common in physics: nearest-neighbor terms on a lattice overlap on only a few sites, so most commutators vanish.
General term-count bounds
For H = sum over gamma of H_gamma, the tight first-order bound involves sum over pairs of ||[H_gamma, H_gamma']||, and higher orders involve nested commutators. Childs, Su, Tran, Wiebe, and Zhu (2021) gave commutator bounds that dramatically improve resource estimates for lattice and chemistry Hamiltonians.
- First-order per-step error: O(t^2) with prefactor from pairwise commutators.
- Order-2k per-step error: O(t^{2k+1}) from higher nested commutators.
- Total error after r steps of size t/r: r times the per-step error at scale t/r.
Empirical scaling and worst case
Because worst-case bounds can be loose, practitioners often measure error numerically on small instances and extrapolate. Randomized orderings of the H_gamma (qDRIFT and random-permutation Trotter) can reduce error further by averaging out systematic bias, trading determinism for better expected accuracy.
Practical takeaways
Pick the step size and order together to hit epsilon with the fewest gates. Exploit locality so that only nearby commutators contribute. When the Hamiltonian has widely varying term norms, consider splitting the large terms into more slices than the small ones. Careful error analysis is what turns Trotterization from a rough heuristic into a method with provable accuracy guarantees, which matters when eigenvalues feed downstream physics.