Magic Phase Transitions in Random Quantum Circuits

In recent years, measurement-induced phase transition, wherein unitary gates and projective measurements compete to determine the entanglement structure of a quantum state, have received intense interest. While entanglement is an important resource in quantum communication, it does not capture the non-classicality needed for quantum computation. Magic refers to the family of measures used to determine how useful the state is for fault-tolerant synthesis of non-Clifford operations in qudit-based error correction architectures — a good magic monotone is zero for a stabilizer state and does not increase under stabilizer operations. In this work, we show that a random stabilizer code subject to coherent errors exhibits a phase transition in magic. Below an error-rate threshold, stabilizer syndrome measurements remove the accumulated magic in the circuit, effectively protecting against the coherent errors. Above the threshold, the syndrome measurements concentrate magic. This "magic" phase transition is intimately related to the error-correction threshold. In this work, we present numerical and analytic characterizations of the magic transition.