Rise and fall of nonstabilizerness via random measurements

TL;DR

This paper studies how nonstabilizerness, or 'magic', evolves in monitored quantum circuits with random Clifford unitaries and measurements, revealing protection mechanisms and conditions for creating or destroying quantum resources.

Contribution

It provides an analytical model for nonstabilizerness dynamics, distinguishing effects of measurement basis and initial states, and highlights differences between coarse and fine-grained diagnostics.

Findings

Nullity decays in quantized steps, requiring exponentially many measurements to vanish.

Measurements in non-Clifford bases can create and sustain nonstabilizerness.

Haar-random states equilibrate quickly, stabilizer states relax linearly with system size.

Abstract

We investigate the dynamics of nonstabilizerness - also known as `magic' - in monitored quantum circuits composed of random Clifford unitaries and local projective 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 non-trivial 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 R\'enyi 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.