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Spectral properties versus magic generation in T-doped random Clifford circuits

Dominik Szombathy1,2,*, Angelo Valli1,3,4, Cătălin Paşcu Moca3,5, János Asbóth1,4,6, Lóránt Farkas2, Tibor Rakovszky1,7, and Gergely Zaránd1,3,†

  • *Contact author: szombathy.dominik@edu.bme.hu
  • †Contact author: zarand.gergely.attila@ttk.bme.hu
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Phys. Rev. Research 7, 043080 – Published 20 October, 2025

DOI: https://doi.org/10.1103/557f-6tpb

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 Rényi entropy distribution and the nonstabilizing 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 nonstabilizing power of the quantum circuit ensemble. In the dilute limit, NT≪N, magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of T gates, NT, and displays a discrete distribution for small NT. At NT≈N, the distribution becomes quasicontinuous, and for NT≫N, it converges to that of Haar-random unitaries, and averages to a finite magic density, m2, limN→∞〈m2〉Haar=1. This is in contrast to the spectral transition, where 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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Phys. Rev. Research 7, 043080– Published 20 October, 2025
  • Received 6 February 2025
  • Accepted 15 September 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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