  A many-body Hamiltonian can be block-diagonalized by expressing it in terms
of symmetry-adapted basis states. Finding the group orbit representatives of
these basis states and their corresponding symmetries is currently a
memory/computational bottleneck on classical computers during exact
diagonalization. We apply Grover's search in the form of a minimization
procedure to solve this problem. Our quantum solution provides an exponential
reduction in memory, and a quadratic speedup in time over classical methods. We
discuss explicitly the full circuit implementation of Grover minimization as
applied to this problem, finding that the oracle only scales as polylog in the
size of the group, which acts as the search space. Further, we design an error
mitigation scheme that, with no additional qubits, reduces the impact of
bit-flip errors on the computation, with the magnitude of mitigation directly
correlated with the error rate, improving the utility of the algorithm in the
Noisy Intermediate Scale Quantum era.
