Benchmarking non-simulable quantum processes via symmetry conservation

Abstract: 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.