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Rise and fall of nonstabilizerness via random measurements

Annarita Scocco1,*, Wai-Keong Mok2,†, Leandro Aolita3, Mario Collura4,5,‡, and Tobias Haug3,§

  • *Contact author: annarita.scocco@unina.it
  • †Contact author: darielmok@caltech.edu
  • ‡Contact author: mcollura@sissa.it
  • §Contact author: tobias.haug@u.nus.edu
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Phys. Rev. Research 8, 013217 – Published 26 February, 2026

DOI: https://doi.org/10.1103/31sq-k4m3

Abstract

We investigate the dynamics of nonstabilizerness—also known as “magic”—in monitored quantum circuits composed of random Clifford unitaries and local measurements. For measurements in the computational basis, we derive an analytical model for dynamics of the stabilizer nullity, showing that it decays in quantized steps and requires exponentially many measurements to vanish, which reveals the strong protection through Clifford scrambling. On the other hand, for measurements performed in rotated non-Clifford bases, measurements can both create and destroy nonstabilizerness. Here, the dynamics leads to a steady state with nontrivial nonstabilizerness, independent of the initial state. We find that Haar-random states equilibrate in constant time, whereas stabilizer states exhibit linear-in-size relaxation time. While the stabilizer nullity is insensitive to the rotation angle, stabilizer Rényi entropies expose a richer structure in their dynamics. Our results uncover sharp distinctions between coarse and fine-grained nonstabilizerness diagnostics and demonstrate how measurements can both suppress and sustain quantum computational resources.

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Outline

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Phys. Rev. Research 8, 013217– Published 26 February, 2026
  • Received 31 July 2025
  • Accepted 23 December 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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