A Bayesian analysis of classical shadows

Abstract

The method of classical shadows heralds unprecedented opportunities for quantum estimation with limited measurements [H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020)]. Yet its relationship to established quantum tomographic approaches, particularly those based on likelihood models, remains unclear. In this article, we investigate classical shadows through the lens of Bayesian mean estimation (BME). In direct tests on numerical data, BME is found to attain significantly lower error on average, but classical shadows prove remarkably more accurate in specific situations -- such as high-fidelity ground truth states -- which are improbable in a fully uniform Hilbert space. We then introduce an observable-oriented pseudo-likelihood that successfully emulates the dimension-independence and state-specific optimality of classical shadows, but within a Bayesian framework that ensures only physical states. Our research reveals how classical shadows effect important departures from conventional thinking in quantum state estimation, as well as the utility of Bayesian methods for uncovering and formalizing statistical assumptions.

arXiv comment8 pages, 5 figures

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Cited by in corpus (11)

• The Variational Quantum Eigensolver: a review of methods and best practices
• Learning Quantum Systems
• Classical Shadows With Noise
• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
• Classical shadows with Pauli-invariant unitary ensembles
• Shadow tomography on general measurement frames
• Bayesian homodyne and heterodyne tomography
• Improving application performance with biased distributions of quantum states
• Data-Centric Machine Learning in Quantum Information Science
• Demonstration of machine-learning-enhanced Bayesian quantum state estimation
• Quantum Circuit Cutting for Classical Shadows