Error Resilience and Unitary 2-Designs

Abstract

Encoding-error-decoding circuits provide a setting for studying protection of quantum information against errors. In these circuits, an encoding unitary is followed by an error layer, decoding by the inverse unitary, and ancilla postselection. For encoders drawn from unitary 2-design ensembles, these circuits exhibit an error-protecting phase below a threshold error strength in which the fidelity between the input and recovered logical states approaches unity. Here, we show how departures from Haar second-moment statistics of the encoder can destabilize the error-protecting phase and derive a universal inequality linking recovery fidelity to the second unitary frame potential, which quantifies the ensemble's proximity to a unitary 2-design. This connection is exemplified by matchgate circuits doped with a tunable number of non-Gaussian gates. We analytically show that undoped circuits lack a stable error-protecting phase in the thermodynamic limit and that increasing the doping restores the error protection mechanism as the encoder ensemble approaches an approximate unitary 2-design. Further examples of Hamiltonian encoders, including an experimentally motivated all-to-all spin model, follow the same mechanism. Our results identify recovery fidelity as a diagnostic of departures from unitary 2-design behavior in quantum many-body systems.