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Statistical learning on randomized data to verify quantum state approximate k-designs

Kaustav Mukherjee1,2,*, Sarah Chehade1,2, Lorenzo Versini3,4, Karim K. Alaa El-Din3,4, Florian Mintert3,5, and Rick Mukherjee1,2,3

  • *Contact author: Kaustav-Mukherjee@utc.edu
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Phys. Rev. Research 8, 023354 – Published 29 June, 2026

DOI: https://doi.org/10.1103/16yg-xqtd

Abstract

Random ensembles of pure states have proven to be extremely important in various aspects of quantum physics such as benchmarking the performance of quantum circuits, testing for quantum advantage, studying many-body thermalization, and the black hole information paradox. Although generating a truly random quantum ensemble is experimentally challenging, approximate realizations are equally valuable and are known to emerge naturally in a variety of physical models, including Rydberg setups. These are referred to as approximate quantum state designs, and verifying their degree of randomness can be a measurement-intensive task, similar to performing full quantum state tomography on many-body systems. In this theoretical work, we present a measurement scheme and analysis techniques to validate the degree of randomness of a quantum ensemble generated by a simulated experimental setup. This is achieved by translating the information residing in the complex many-body state into a succinct representation of classical data using projective measurements in randomly chosen bases, which is then processed using methods of statistical inference such as maximum-likelihood estimation and neural networks, benchmarked against the predictions of shadow tomography. Our scheme only requires individually addressed single-qubit operations to be performed in order to be employed, making it applicable for a range of physical platforms.

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Phys. Rev. Research 8, 023354– Published 29 June, 2026
  • Received 3 November 2025
  • Accepted 4 June 2026
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