# Predicting adaptively chosen observables in quantum systems

arXiv: https://arxiv.org/abs/2410.15501 · HTML: https://mlsanity.com/paper/2410.15501/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 82 references that are in the map.
- First posted on arXiv 20 Oct 2024.

## Builds on (most cited first, 17 of 17)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Algorithmic Stability for Adaptive Data Analysis](https://mlsanity.com/paper/1511.02513/) (2015)
- [Generalization in Adaptive Data Analysis and Holdout Reuse](https://mlsanity.com/paper/1506.02629/) (2015)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [A New Analysis of Differential Privacy's Generalization Guarantees](https://mlsanity.com/paper/1909.03577/) (2019)
- [Interactive Fingerprinting Codes and the Hardness of Preventing False Discovery](https://mlsanity.com/paper/1410.1228/) (2014)
- [Calibrating Noise to Variance in Adaptive Data Analysis](https://mlsanity.com/paper/1712.07196/) (2017)
- [Quantum SDP-Solvers: Better upper and lower bounds](https://mlsanity.com/paper/1705.01843/) (2017)
- [Generalization for Adaptively-chosen Estimators via Stable Median](https://mlsanity.com/paper/1706.05069/) (2017)
- [Efficient estimation of Pauli observables by derandomization](https://mlsanity.com/paper/2103.07510/) (2021)
- [A bounded-noise mechanism for differential privacy](https://mlsanity.com/paper/2012.03817/) (2020)
- [Classical Shadows With Noise](https://mlsanity.com/paper/2011.11580/) (2020)
- [Subsampling Suffices for Adaptive Data Analysis](https://mlsanity.com/paper/2302.08661/) (2023)
- [On Avoiding the Union Bound When Answering Multiple Differentially Private Queries](https://mlsanity.com/paper/2012.09116/) (2020)
- [Privately Answering Counting Queries with Generalized Gaussian Mechanisms](https://mlsanity.com/paper/2010.01457/) (2020)
- [Dealing with Range Anxiety in Mean Estimation via Statistical Queries](https://mlsanity.com/paper/1611.06475/) (2016)

## Abstract (arXiv)

Recent advances have demonstrated that O(log M) 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 Omega(√M) samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of M adaptively chosen local and Pauli observables. 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(log M) samples, independent of system size. Our results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide new algorithmic tools to safeguard against erroneous predictions in quantum experiments.