Spectral properties versus magic generation in T -doped random Clifford circuits

We study the emergence of complexity in deep random N-qubit T-gate-doped Clifford circuits, as reflected in their spectral properties and in magic generation, characterized by the stabilizer Rényi entropy distribution and the nonstabilizing power of the circuit. For pure (undoped) Clifford circuits, a unique periodic orbit structure in the space of Pauli strings implies peculiar spectral correlations and level statistics with large degeneracies. T-gate doping induces an exponentially fast transition to chaotic behavior, described by random matrix theory. We compare these complexity indicators with magic generation properties of the Clifford +T ensemble, and determine the distribution of magic, as well as the average nonstabilizing power of the quantum circuit ensemble. In the dilute limit, NT≪N, magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of T gates, NT, and displays a discrete distribution for small NT. At NT≈N, the distribution becomes quasicontinuous, and for NT≫N, it converges to that of Haar-random unitaries, and averages to a finite magic density, m2, limN→∞〈m2〉Haar=1. This is in contrast to the spectral transition, where O(1) T gates suffice to remove spectral degeneracies and to induce a transition to chaotic behavior in the thermodynamic limit. Magic is therefore a more sensitive indicator of complexity.