  Recent work has deployed linear combinations of unitaries techniques to
reduce the cost of fault-tolerant quantum simulations of correlated electron
models. Here, we show that one can sometimes improve upon those results with
optimized implementations of Trotter-Suzuki-based product formulas. We show
that low-order Trotter methods perform surprisingly well when used with phase
estimation to compute relative precision quantities (e.g. energies per unit
cell), as is often the goal for condensed-phase systems. In this context,
simulations of the Hubbard and plane-wave electronic structure models with $N <
10^5$ fermionic modes can be performed with roughly $O(1)$ and $O(N^2)$ T
complexities. We perform numerics revealing tradeoffs between the error and
gate complexity of a Trotter step; e.g., we show that split-operator techniques
have less Trotter error than popular alternatives. By compiling to surface code
fault-tolerant gates and assuming error rates of one part per thousand, we show
that one can error-correct quantum simulations of interesting, classically
intractable instances with a few hundred thousand physical qubits.
