# Advantage of Sample Complexity in Quantum PAC Learning Requires Inverse Access to State-Preparation Unitaries

arXiv: https://arxiv.org/abs/2609.38403 · HTML: https://mlsanity.com/paper/2609.38403/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 37 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 8 of 8)

- [The Optimal Sample Complexity of PAC Learning](https://mlsanity.com/paper/1507.00473/) (2015)
- [Optimal Quantum Sample Complexity of Learning Algorithms](https://mlsanity.com/paper/1607.00932/) (2016)
- [Exponential separations between classical and quantum learners](https://mlsanity.com/paper/2306.16028/) (2023)
- [On establishing learning separations between classical and quantum machine learning with classical data](https://mlsanity.com/paper/2208.06339/) (2022)
- [Quantum tomography using state-preparation unitaries](https://mlsanity.com/paper/2207.08800/) (2022)
- [Amplitude amplification and estimation require inverses](https://mlsanity.com/paper/2507.23787/) (2025)
- [Advantage of Quantum Machine Learning from General Computational Advantages](https://mlsanity.com/paper/2312.03057/) (2023)
- [Are controlled unitaries helpful?](https://mlsanity.com/paper/2508.00055/) (2025)

## Abstract (arXiv)

Whether quantum computation can reduce the amount of data sampled from an unknown probability distribution required to learn a prediction rule is a fundamental question in quantum machine learning. Quantum PAC learning studies this question using quantum data as a quantum state whose squared amplitudes encode the unknown distribution from which classical learning data are sampled. With only copies of such quantum data, the optimal worst-case sample complexity asymptotically matches that of classical PAC learning. In contrast, access to both a state-preparation unitary for this state and its inverse can improve the query-complexity dependence on the accuracy parameter in realizable learning. However, it has remained unclear whether forward-only access allows such an improvement. In this work, taking the worst case over compatible state-preparation unitaries and their finite ambient dimensions, we show that the optimal forward-only query complexities of realizable and agnostic learning are, respectively, Theta((d+log(1/δ))/varepsilon) and Theta((d+log(1/δ))/varepsilon2), where d is the VC dimension of the concept class, varepsilon the accuracy parameter, and δ the failure probability. These bounds match the optimal sample complexities with classical data or quantum data copies. To prove them, we establish a reduction using q copies of the prepared state to approximate the Haar-averaged output of any q-query forward-only algorithm. These results show that forward-only access cannot provide an asymptotic query-complexity advantage over learning from classical data or quantum data copies in this worst-case setting, and establish the essential role of inverse access in the known realizable-setting improvement. Our reduction also provides a new framework for analyzing limitations of forward state-preparation access via state-copy lower bounds.