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 non-stabilizing 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 non-stabilizing power of the quantum circuit ensemble. In the dilute limit, N_T ll N, magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of T-gates, N_T, and displays a discrete distribution for small N_T. At N_Tapprox N, the distribution becomes quasi-continuous, and for N_Tgg N it converges to that of Haar-random unitaries, and averages to a finite magic density, m₂, lim_N→infty langle m₂ rangle_Haar = 1. This is in contrast to the spectral transition, where {cal 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.