Hamiltonian simulation is one of the most promising candidates for the demonstration of quantum advantage within the next ten years, and several studies have proposed end-to-end resource estimates for executing such algorithms on fault-tolerant quantum processors. Usually, these resource estimates are based upon the assumption that quantum error correction is implemented using the surface code, and that the best surface code compilation scheme involves serializing input circuits by eliminating all Clifford gates. This transformation is thought to make best use of the native multi-body measurement (lattice surgery) instruction set available to surface codes. Some work, however, has suggested that direct compilation from Clifford+T to lattice surgery operations may be beneficial for circuits that have high degrees of logical parallelism. In this study, we analyze the resource costs for implementing Hamiltonian simulation using example approaches from each of these leading surface code compilation families. The Hamiltonians whose dynamics we consider are those of the transverse-field Ising model in several geometries, the Kitaev honeycomb model, and the $\mathrm{\alpha-RuCl_3}$ complex under a time-varying magnetic field. We show, among other things, that the optimal scheme depends on whether Hamiltonian simulation is implemented using the quantum signal processing or Trotter-Suzuki algorithms, with Trotterization benefiting by orders of magnitude from direct Clifford+T compilation for these applications. Our results suggest that surface code quantum computers should not have a one-size-fits-all compilation scheme, but that smart compilers should predict the optimal scheme based upon high-level quantities from logical circuits such as average circuit density, numbers of logical qubits, and T fraction.
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