Higher-Order Trotter Formulas
Symmetric and recursively constructed product formulas that cancel low-order error terms for far better accuracy per step.
The symmetric second-order formula
The first-order formula has an error of order t^2 per step. Symmetrizing it removes the leading term. The second-order Strang splitting is S2(t) = e^(-iA t/2) e^(-iB t) e^(-iA t/2). Because it is time-symmetric, S2(t) S2(-t) = I, and its per-step error is O(t^3).
Suzuki's recursion
Masuo Suzuki showed how to build arbitrarily high-order formulas from S2. The order-2k formula is defined recursively: S_{2k}(t) = S_{2k-2}(p_k t)^2 S_{2k-2}((1 - 4 p_k) t) S_{2k-2}(p_k t)^2, with p_k = 1 / (4 - 4^{1/(2k-1)}). Each level cancels the next error term, giving per-step error O(t^{2k+1}).
The cost trade
Higher order means fewer, larger steps for a target accuracy, but each step contains exponentially more exponentials in the order (roughly 5^(k-1) copies of S2). There is an optimal order for any given t and epsilon. Rough scaling: the number of gates for a 2k-order formula behaves like (t^{1 + 1/(2k)}) times a factor growing with 5^k, so moderate orders (2 to 6) are usually best.
- Order 1: error O(t^2), simplest, most gates for high precision.
- Order 2 (Strang): error O(t^3), a common practical choice.
- Order 4 and higher: better asymptotic scaling in epsilon, larger constant factors.
- Choose order by minimizing total gate count for the specific t and epsilon.
Commutator-aware bounds
Modern analyses (Childs and collaborators) show that Trotter error depends on nested commutators of the H_j, not on their norms alone. For Hamiltonians with local, mostly commuting structure the true error is much smaller than naive bounds, which makes even modest-order formulas competitive with post-Trotter methods for many physical systems.
Higher-order product formulas are the practical bridge: they keep the zero-ancilla simplicity of Trotterization while pushing the error scaling toward what more elaborate algorithms achieve.