  Simulating the dynamics of quantum systems is an important application of
quantum computers and has seen a variety of implementations on current
hardware. We show that by introducing quantum gates implementing unitary
transformations generated by the symmetries of the system, one can induce
destructive interference between the errors from different steps of the
simulation, effectively giving faster quantum simulation by symmetry
protection. We derive rigorous bounds on the error of a symmetry-protected
simulation algorithm and identify conditions for optimal symmetry protection.
In particular, when the symmetry transformations are chosen as powers of a
unitary, the error of the algorithm is approximately projected to the so-called
quantum Zeno subspaces. We prove a bound on this approximation error,
exponentially improving a recent result of Burgarth, Facchi, Gramegna, and
Pascazio. We apply the symmetry protection technique to the simulations of the
XXZ Heisenberg interactions with local disorder and the Schwinger model in
quantum field theory. For both systems, the technique can reduce the simulation
error by several orders of magnitude over the unprotected simulation. Finally,
we provide numerical evidence suggesting that the technique can also protect
simulation against other types of coherent, temporally correlated errors, such
as the $1/f$ noise commonly found in solid-state experiments.
