  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. In this work, we develop quantum algorithms for Hamiltonian
simulation "one level below" the circuit model, exploiting the underlying
control over qubit interactions available in principle in most quantum hardware
implementations. We then analyse the impact of these techniques under the
standard error model where errors occur per gate, and an error model with a
constant error rate per unit time.
  To quantify the benefits of this approach, we apply it to a canonical
example: time-dynamics simulation of the 2D spin Fermi-Hubbard model. We derive
analytic circuit identities for efficiently synthesising multi-qubit evolutions
from two-qubit interactions. Combined with new error bounds for Trotter product
formulas tailored to the non-asymptotic regime and a careful analysis of error
propagation under the aforementioned per-gate and per-time error models, we
improve upon the previous best methods for Hamiltonian simulation by multiple
orders of magnitude. By our calculations, for a 5$\mathbf\times$5 Fermi-Hubbard
lattice we reduce the circuit depth from 800,160 to 1460 in the per-gate error
model, or the circuit-depth-equivalent to 440 in the per-time error model. This
brings Hamiltonian simulation, previously beyond reach of current hardware for
non-trivial examples, significantly closer to being feasible in the NISQ era.
