  Quantum circuits with local particle number conservation (LPNC) restrict the
quantum computation to a subspace of the Hilbert space of the qubit register.
In a noiseless or fault-tolerant quantum computation, such quantities are
preserved. In the presence of noise, however, the evolution's symmetry could be
broken and non-valid states could be sampled at the end of the computation. On
the other hand, the restriction to a subspace in the ideal case suggest the
possibility of more resource efficient error mitigation techniques for circuits
preserving symmetries that are not possible for general circuits. Here, we
analyze the probability of staying in such symmetry-preserved subspaces under
noise, providing an exact formula for local depolarizing noise. We apply our
findings to benchmark, under depolarizing noise, the symmetry robustness of
XY-QAOA, which has local particle number conserving symmetries, and is a
special case of the Quantum Alternating Operator Ansatz. We also analyze the
influence of the choice of encoding the problem on the symmetry robustness of
the algorithm and discuss a simple adaption of the bit flip code to correct for
symmetry-breaking errors with reduced resources.
