The quantum circuit model is the de-facto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In the era of near-term, noisy, intermediate-scale quantum (NISQ) hardware with severely restricted resources, this overhead may be unjustifiable. We develop quantum algorithms for Hamiltonian simulation "one level below" the circuit model, exploiting the underlying control over qubit interactions available in most quantum hardware implementations. To quantify the benefits, we apply this to a canonical example: time-dynamics simulation of the 2D spin Fermi-Hubbard model.
Using new error bounds for Trotter product formulas tailored to the non-asymptotic regime and an analysis of error propagation in this "sub-circuit" model, we improve upon the previous best methods for Hamiltonian simulation by multiple orders of magnitude. E.g. for a 5×5 Fermi-Hubbard lattice, we reduce the circuit depth from 800,160 to 440 in the best case. This brings Hamiltonian simulation, previously beyond reach of current hardware for non-trivial examples, significantly closer to being feasible in the NISQ era.