Spectral Properties and Magic Generation in \(T\)-doped Random Clifford Circuits
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| Publicado no: | arXiv.org (Dec 20, 2024), p. n/a |
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| Autor principal: | |
| Outros Autores: | , , , , , |
| Publicado em: |
Cornell University Library, arXiv.org
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| Acesso em linha: | Citation/Abstract Full text outside of ProQuest |
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| Resumo: | 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. 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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| ISSN: | 2331-8422 |
| Fonte: | Engineering Database |