Spectral Properties Versus Magic Generation in $T$-doped Random Clifford Circuits
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2024
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| author | Szombathy, Dominik Valli, Angelo Moca, Cătălin Paşcu Asbóth, János Farkas, Lóránt Rakovszky, Tibor Zaránd, Gergely |
| author_facet | Szombathy, Dominik Valli, Angelo Moca, Cătălin Paşcu Asbóth, János Farkas, Lóránt Rakovszky, Tibor Zaránd, Gergely |
| contents | 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 $N_T\approx N$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_15912 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Properties Versus Magic Generation in $T$-doped Random Clifford Circuits Szombathy, Dominik Valli, Angelo Moca, Cătălin Paşcu Asbóth, János Farkas, Lóránt Rakovszky, Tibor Zaránd, Gergely Quantum Physics 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 $N_T\approx N$, 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. |
| title | Spectral Properties Versus Magic Generation in $T$-doped Random Clifford Circuits |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2412.15912 |