Random unitaries with SU(d) Symmetry: OTOCs and Covariant Codes

Quantum information processing in continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. Continuous symmetry is always associated with some conserved quantity from Noether's theorem on which much modern understanding of physics is built. In recent years, continuous symmetries have drawn considerable interest in quantum information, ranging from quantum hydrodynamics, quantum error correction, quantum communication, and quantum machine learning. Central to many applications is the ability that local interactions can still lead to sufficient randomness in the presence of continuous symmetry, which is characterized by the notion of Harr randomness in the late-time using tools such as the out-of-time-ordered correlator (OTOC) and entanglement. For instance, the late-time OTOC value with respect to a single site for Haar random circuits is isotropic and decrease exponentially in finite-size circuit, which can be understood classically. Surprisingly, charge conservation on random circuits could drastically change the circuit behavior and often renders power-law decay in OTOCs in the finite-size circuit, and an outstanding question is the behavior of random circuits under non-Abelian charges—for which SU(d) symmetry provides a canonical example. We provide the first step towards this direction by showing the under SU(d) conservation law, the finite-size OTOC with respect to a given site decays in on qubits. Based on this result, we further explore applications of random SU(d)-symmetric circuits in approximate covariant quantum error correction, where we show that random SU(d)-symmetric unitaries with constant encoding are nearly optimal covariant codes, saturating the fundamental limit imposed by the approximate Eastin-Knill theorem. Our work invites further research on quantum information with continuous symmetries, where the mathematical tools developed in this work are expected to be useful.