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Predicting Adaptively Chosen Observables in Quantum Systems

Huang, Jerry [California Institute of Technology (CalTech), Pasadena, CA (United States)]ยทLewis, Laura [California Institute of Technology (CalTech), Pasadena, CA (United States); Google Quantum AI, Venice, CA (United States); University of Cambridge (United Kingdom)] (ORCID:0000000177938345)ยทHuang, Hsin-Yuan [California Institute of Technology (CalTech), Pasadena, CA (United States); Google Quantum AI, Venice, CA (United States); Massachusetts Institute of Technology, Cambridge, MA (United States)] (ORCID:0000000153172613)ยทPreskill, John [California Institute of Technology (CalTech), Pasadena, CA (United States); AWS Center for Quantum Computing, Pasadena, CA (United States)]
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Abstract

Recent advances have demonstrated that ๐’ชโก(log ๐‘€) measurements suffice to predict ๐‘€ 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 ฮฉโก(โˆš๐‘€) samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of ๐‘€ adaptively chosen local and Pauli observables, where the system size scales exponentially and polynomially in ๐‘€, 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 ๐’ชโก(log ๐‘€) 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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Huang, Jerry [California Institute of Technology (CalTech), Pasadena, CA (United States)], Lewis, Laura [California Institute of Technology (CalTech), Pasadena, CA (United States); Google Quantum AI, Venice, CA (United States); University of Cambridge (United Kingdom)] (ORCID:0000000177938345), Huang, Hsin-Yuan [California Institute of Technology (CalTech), Pasadena, CA (United States); Google Quantum AI, Venice, CA (United States); Massachusetts Institute of Technology, Cambridge, MA (United States)] (ORCID:0000000153172613), Preskill, John [California Institute of Technology (CalTech), Pasadena, CA (United States); AWS Center for Quantum Computing, Pasadena, CA (United States)]. 2026-03-09. Predicting Adaptively Chosen Observables in Quantum Systems. https://doi.org/10.1103/xhn1-vnp9

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