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  • Open Access

Predicting Adaptively Chosen Observables in Quantum Systems

Jerry Huang1,*, Laura Lewis1,2,3, Hsin-Yuan Huang1,2,4, and John Preskill1,5

  • *Contact author: jyhuang@alumni.caltech.edu
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PRX Quantum 7, 010347 – Published 9 March, 2026

DOI: https://doi.org/10.1103/xhn1-vnp9

Abstract

Recent advances have demonstrated that O(logM) measurements suffice to predict M properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are chosen independently of the data. This assumption can be violated in practice, where scientists adaptively select properties after looking at previous predictions. This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that Ω(M) samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of M adaptively chosen local and Pauli observables, where the system size scales exponentially and polynomially in M, respectively. We also present computationally efficient algorithms that achieve this information-theoretic lower bound. In contrast, for bounded-Frobenius-norm observables, we devise an algorithm requiring only O(logM) samples, independent of system size. These results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide algorithmic tools to safeguard against erroneous predictions in quantum experiments.

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Popular Summary

Learning properties of quantum systems from experimental data constitutes a fundamental pillar of quantum information science. Recent breakthroughs have shown that one can accurately predict many properties of a quantum state using surprisingly few measurements—only growing logarithmically with the number of properties. These results raise the hope that large quantum systems can be efficiently characterized using minimal quantum data.

However, these guarantees crucially depend on the assumption that properties are chosen nonadaptively—a constraint that stands in tension with the inherently adaptive nature of scientific inquiry, where hypotheses are often tested and revised. Our work shows that adaptivity can in fact undermine these guarantees. For two key classes of quantum measurements—local and Pauli observables—we prove that the number of required measurements scales polynomially with the number of properties, much larger than the logarithmic scaling in the nonadaptive case. We also design algorithms that achieve this optimal scaling, showing that the limit is fundamental. In contrast, for a third physically relevant class of measurements (bounded-Frobenius-norm observables), we show only a logarithmic number of samples are required, recovering the original efficiency.

Our results reveal that adaptivity, while essential for science, can undermine the statistical reliability of quantum predictions. We also provide algorithms that protect against these pitfalls. Our findings pave the way toward identifying conditions under which quantum data can be reused and developing robust, efficient quantum algorithms for scientific discovery.

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PRX Quantum 7, 010347– Published 9 March, 2026
  • Received 12 June 2025
  • Accepted 28 January 2026
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