Near-optimal covariant quantum error-correcting codes from random unitaries with symmetries

Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes that obey (are covariant with respect to) continuous symmetries in a certain sense cannot correct erasure errors perfectly (a well-known result in this regard being the Eastin-Knill theorem in the context of fault-tolerant quantum computing), in contrast to the case without symmetry constraints. Furthermore, several quantitative fundamental limits on the accuracy of such covariant codes for approximate quantum error correction are known. Here, we consider the quantum error correction capability of uniformly random covariant codes. In particular, we analytically study the most essential cases of $U(1)$ and $SU(d)$ symmetries, and show that for both symmetry groups the error of the covariant codes generated by Haar-random symmetric unitaries, i.e., unitaries that commute with the group actions, typically scale as $O(n^{-1})$ in terms of both the average- and worst-case purified distances against erasure noise, saturating the fundamental limits to leading order. We note that the results hold for symmetric variants of unitary 2-designs, and comment on the convergence problem of symmetric random circuits. Our results not only indicate (potentially efficient) randomized constructions of optimal $U(1)$- and $SU(d)$-covariant codes, but also reveal fundamental properties of random symmetric unitaries, which yield important solvable models of complex quantum systems (including black holes and many-body spin systems) that have attracted great recent interest in quantum gravity and condensed matter physics. We expect our construction and analysis to find broad relevance in both physics and quantum computing.