  The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed
for the study of quantum gauge field theories. We give scalable, explicit
digital quantum algorithms to simulate the lattice Schwinger model in both NISQ
and fault-tolerant settings. In particular, we perform a tight analysis of
low-order Trotter formula simulations of the Schwinger model, using recently
derived commutator bounds, and give upper bounds on the resources needed for
simulations in both scenarios. In lattice units, we find a Schwinger model on
$N/2$ physical sites with coupling constant $x^{-1/2}$ and electric field
cutoff $x^{-1/2}\Lambda$ can be simulated on a quantum computer for time $2xT$
using a number of $T$-gates or CNOTs in $\widetilde{O}( N^{3/2} T^{3/2}
\sqrt{x} \Lambda )$ for fixed operator error. This scaling with the truncation
$\Lambda$ is better than that expected from algorithms such as qubitization or
QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to
estimate observables which we cost in both the NISQ and fault-tolerant settings
by assuming a simple target observable---the mean pair density. Finally, we
bound the root-mean-square error in estimating this observable via simulation
as a function of the diamond distance between the ideal and actual CNOT
channels. This work provides a rigorous analysis of simulating the Schwinger
model, while also providing benchmarks against which subsequent simulation
algorithms can be tested.
