Dynamic simulation of materials is a promising application for near-term
quantum computers. Current algorithms for Hamiltonian simulation, however,
produce circuits that grow in depth with increasing simulation time, limiting
feasible simulations to short-time dynamics. Here, we present a method for
generating circuits that are constant in depth with increasing simulation time
for a subset of one-dimensional materials Hamiltonians, thereby enabling
simulations out to arbitrarily long times. Furthermore, by removing the
effective limit on the number of feasibly simulatable time-steps, the
constant-depth circuits enable Trotter error to be made negligibly small by
allowing simulations to be broken into arbitrarily many time-steps. Composed of
two-qubit matchgates on nearest-neighbor qubits, these constant-depth circuits
are constructed based on a set of multi-matchgate identity relationships. For
an $N$-spin system, the constant-depth circuit contains only $\mathcal{O}(N^2)$
CNOT gates. When compared to standard Hamiltonian simulation algorithms, our
method generates circuits with order-of-magnitude fewer gates, which allows us
to successfully simulate the long-time dynamics of systems with up to 5 spins
on available quantum hardware. This paves the way for simulations of long-time
dynamics for scientifically and technologically relevant quantum materials,
enabling the observation of interesting and important atomic-level physics.