# A Theoretical Framework for Learning from Quantum Data

arXiv: https://arxiv.org/abs/2107.06406 · HTML: https://mlsanity.com/paper/2107.06406/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jul 2021.

## Built on by (most cited first, 8 of 8)

- [Sample Complexity of Learning Parametric Quantum Circuits](https://mlsanity.com/paper/2107.09078/) (2021)
- [Toward Physically Realizable Quantum Neural Networks](https://mlsanity.com/paper/2203.12092/) (2022)
- [Efficient Gradient Estimation for Parameterized Quantum Systems with Lie Algebraic Symmetries](https://mlsanity.com/paper/2404.05108/) (2024)
- [Unclonability and Quantum Cryptanalysis: From Foundations to Applications](https://mlsanity.com/paper/2210.17545/) (2022)
- [Fat Shattering, Joint Measurability, and PAC Learnability of POVM Hypothesis Classes](https://mlsanity.com/paper/2308.12304/) (2023)
- [New Bounds on Quantum Sample Complexity of Measurement Classes](https://mlsanity.com/paper/2408.12683/) (2024)
- [Qute: Towards Quantum-Native Database](https://mlsanity.com/paper/2602.14699/) (2026)
- [Advantage of Entangled Learning Rules in Quantum Measurement Class Learning](https://mlsanity.com/paper/2610.07328/) (2026)

## Abstract (arXiv)

Over decades traditional information theory of source and channel coding advances toward learning and effective extraction of information from data. We propose to go one step further and offer a theoretical foundation for learning classical patterns from quantum data. However, there are several roadblocks to lay the groundwork for such a generalization. First, classical data must be replaced by a density operator over a Hilbert space. Hence, deviated from problems such as state tomography, our samples are i.i.d density operators. The second challenge is even more profound since we must realize that our only interaction with a quantum state is through a measurement which -- due to no-cloning quantum postulate -- loses information after measuring it. With this in mind, we present a quantum counterpart of the well-known PAC framework. Based on that, we propose a quantum analogous of the ERM algorithm for learning measurement hypothesis classes. Then, we establish upper bounds on the quantum sample complexity quantum concept classes.