Spectral Properties and Magic Generation in $T$-doped Random Clifford Circuits

• URL: http://arxiv.org/abs/2412.15912v1
• Date: Fri, 20 Dec 2024 14:04:51 GMT
• Title: Spectral Properties and Magic Generation in $T$-doped Random Clifford Circuits
• Authors: Dominik Szombathy, Angelo Valli, Cătălin Paşcu Moca, János Asbóth, Lóránt Farkas, Tibor Rakovszky, Gergely Zaránd,
• Abstract summary: We study the emergence of complexity in deep random $N$-qubit $T$-gate doped Clifford circuits.<n>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.
• Score: 1.1517315048749441
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 R\'enyi entropy. 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. To characterize magic generation properties of the Clifford+$T$ ensemble, we determine the distribution of magic, as well as the average nonstabilizing power of the quantum circuit ensemble. In the dilute limit, $N_T \ll N$, magic generation is governed by single-qubit behavior, and magic increases linearly with the number of $T$-gates, $N_T$. For $N_T\gg N$, magic distribution converges to that of Haar-random unitaries, and averages to a finite magic density, $\mu$, $\lim_{N\to\infty} \langle\mu\rangle_\text{Haar} = 1$. Although our numerics has large finite-size effects, finite size scaling reveals a magic density phase transition at a critical $T$-gate density, $n^{*}_T = (N_T/N)^* \approx 2.41$ in the $N \to \infty$ limit. 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.

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