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

• URL: http://arxiv.org/abs/2412.15912v2
• Date: Mon, 03 Feb 2025 15:21:20 GMT
• Title: Spectral Properties Versus 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. 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:
• 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 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 $N_T\approx N$, the distribution becomes quasi-continuous, and for $N_T\gg N$ it converges to that of Haar-random unitaries, and averages to a finite magic density, $m_2$, $\lim_{N\to\infty} \langle m_2 \rangle_\text{Haar} = 1$. 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.

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