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  • Open Access

Estimating Nonstabilizerness Dynamics Without Simulating It

Alessio Paviglianiti1,*, Guglielmo Lami1,2, Mario Collura1,3, and Alessandro Silva1

  • *Contact author: paviglianitialessio@gmail.com
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PRX Quantum 6, 030320 – Published 4 August, 2025

DOI: https://doi.org/10.1103/msm2-vmg7

Abstract

We introduce the iterative Clifford circuit renormalization (ICCR), a novel technique designed to handle the dynamics of nonstabilizerness (also known as quantum magic) in generic quantum circuits composed of Clifford and non-Clifford gates, as well as measurements, extending beyond the reach of pre-existing methods. ICCR iteratively adjusts the starting circuit, transforming it into a Clifford circuit where all elements that can alter the nonstabilizerness, such as measurements or T gates, have been removed. In the process the initial state is renormalized in such a way that the new circuit outputs the same final state as the original one. This approach embeds the complex dynamics of nonstabilizerness in the flow of an effective initial state, enabling the efficient evaluation of stabilizer Renyi entropies and magic nullity while avoiding the need for direct and computationally expensive simulation of the original circuit. The initial state renormalization can be computed explicitly using a matrix-product state approximation that can be systematically improved. We implement the ICCR algorithm to evaluate the nonstabilizerness dynamics for systems of size up to N=1000, in one and more dimensions. We validate our method by comparing it to tensor-network simulations. Finally, we employ the ICCR technique to study measurement-induced magic transitions in monitored circuits without and with T gates.

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Popular Summary

Quantum computers are attracting increasing attention because they promise to solve complex problems that are beyond the reach of classical devices by harnessing unique features of quantum mechanics as resources. One such resource, called nonstabilizerness, has recently emerged as a key quantity in quantum physics, constituting a necessary ingredient for quantum advantage. In fact, studying nonstabilizerness not only sheds light on how quantum systems behave, but also quantifies how difficult they are to simulate on classical computers, thus making its investigation both fundamentally important and practically relevant.

In our work, we introduce a new technique to evaluate this resource using classical computers for a broad class of systems known as quantum circuits, which are the quantum counterparts of traditional algorithms. Our approach relies on efficient manipulations of the circuit structure, allowing us to effectively bypass the step of explicitly simulating quantum states, which is typically a major bottleneck. This enables a computationally cheap evaluation of nonstabilizerness and significantly expands the reach of existing methods.

The technique we propose offers a powerful and flexible tool for future research on computational complexity in quantum circuits. On the theoretical side, it can be used to explore quantum dynamics and nonequilibrium phases of matter. On the practical side, it enables the estimation of the computational cost of quantum algorithms and thus the potential for quantum advantage.

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PRX Quantum 6, 030320– Published 4 August, 2025
  • Received 7 December 2024
  • Revised 18 March 2025
  • Accepted 2 July 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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