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  • Letter
  • Open Access

Biased estimator channels for classical shadows

Zhenyu Cai1,2, Adrian Chapman1,*, Hamza Jnane1,2, and Bálint Koczor3,1,2,†

  • *Contact author: adrian.chapman@materials.ox.ac.uk
  • †Contact author: koczor@maths.ox.ac.uk
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Phys. Rev. A 111, L030402 – Published 21 March, 2025

DOI: https://doi.org/10.1103/PhysRevA.111.L030402

Abstract

Extracting classical information from quantum systems is of fundamental importance, and classical shadows allow us to extract a large amount of information using relatively few measurements. Conventional shadow estimators are unbiased and thus agree with the true mean in expectation. In this Letter, we consider a biased scheme, intentionally introducing a bias in the expectation value by rescaling the conventional classical-shadow estimators to reduce the error in the finite-sample regime. The approach is straightforward to implement and requires no quantum resources. We analytically prove average-case as well as worst- and best-case scenarios, and rigorously prove that it is, in principle, always worth biasing the estimators. We illustrate our approach in a quantum simulation task of a 12-qubit spin-ring problem and demonstrate how estimating expected values of nonlocal perturbations can be significantly more efficient using our biased scheme.

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Phys. Rev. A 111, L030402– Published 21 March, 2025
  • Received 1 March 2024
  • Accepted 26 February 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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