  As quantum devices scale up, many-body quantum gates and algorithms begin to
surpass what is possible to simulate classically. Validation methods which rely
on such classical simulation, such as process tomography and randomized
benchmarking, cannot efficiently check correctness of most of the processes
involved. In particular non-Clifford gates are a requirement for not only
universal quantum computation but for any algorithm or quantum simulation that
yields fundamental speedup in comparison with its classical counterpart. We
show that it is in fact still possible to validate such non-simulable processes
by taking advantage of expected or engineered conservations laws in the system,
combined with a unitary one-design strategy to randomize errors over the
computational Hilbert space. We show that in the context of (fault-tolerant)
quantum error correction, we can construct a one-design using the logically
encoded Clifford group over the engineered error-free stabilizer subspace to
obtain average error for arbitrary logically-encoded gates and algorithms. In
the case of benchmarking simulation of physical systems, these can have various
exotic symmetries over which one-design strategies can nonetheless be
constructed. We give examples for fermionic systems which conserve particle
number, as well as for the Fermi-Hubbard model. The symmetry benchmarking
method preserves robustness to state preparation and measurement imperfections.
