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Predicting Magic from Very Few Measurements

J. M. Varela1,2, L. L. Keller2, A. de Oliveira Junior3, D. A. Moreira4, R. Chaves2,4, and R. A. Macêdo1,2,5

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PRX Quantum 7, 043009 – Published 14 October, 2026

DOI: https://doi.org/10.1103/2xnf-jcyn

Abstract

The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using any set of mn-qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements m and polynomial in the number of qubits n. By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless P=NP, no algorithm polynomial in m can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.

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Popular Summary

Quantum computers have the potential to solve certain problems that are beyond the reach of classical computers. A central ingredient behind this advantage is a quantum property known as “magic.” However, detecting it in a quantum system has traditionally been demanding as it requires reconstructing the system’s entire quantum state, with a number of measurements growing exponentially as the system gets larger. In this work, we show that this major obstacle can be overcome by developing a method that can characterize magic with a polynomial number of measurements. We also identify the fundamental computational limits of this approach, showing when determining magic becomes intrinsically difficult. Our results are also practical: For physically relevant quantum states, including low-energy and finite-temperature states of quantum systems, we show that magic can be detected using only a small, fixed number of measurements. This opens up a practical route for identifying magic in quantum devices.

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PRX Quantum 7, 043009– Published 14 October, 2026
  • Received 14 March 2026
  • Revised 1 July 2026
  • Accepted 31 July 2026
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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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