# Entanglement as a Structural Complexity Axis: A PAC-Bayesian View of Generalization in Quantum Policies and Value Functions

arXiv: https://arxiv.org/abs/2607.06230 · HTML: https://mlsanity.com/paper/2607.06230/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Quantum reinforcement learning, one of 15 directions in Deep reinforcement learning methods.
- Builds on 10 of its 13 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Placed in

- Deep reinforcement learning methods / [Quantum reinforcement learning](https://mlsanity.com/deeprl/quantum-reinforcement-learning/)

## Builds on (most cited first, 10 of 10)

- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Quantum reinforcement learning in continuous action space](https://mlsanity.com/paper/2012.10711/) (2020)
- [PAC-Bayes Control: Learning Policies that Provably Generalize to Novel Environments](https://mlsanity.com/paper/1806.04225/) (2018)
- [VQC-Based Reinforcement Learning with Data Re-uploading: Performance and Trainability](https://mlsanity.com/paper/2401.11555/) (2024)
- [Quantum Reinforcement Learning by Adaptive Non-local Observables](https://mlsanity.com/paper/2507.19629/) (2025)
- [A PAC-Bayesian approach to generalization for quantum models](https://mlsanity.com/paper/2603.22964/) (2026)
- [PAC-Bayesian Reinforcement Learning Trains Generalizable Policies](https://mlsanity.com/paper/2510.10544/) (2025)

## Abstract (arXiv)

Parameterized quantum circuits (PQCs) are increasingly used as policies and value functions in quantum reinforcement learning, yet it remains unclear when and why quantum policies generalize. We give a PAC-Bayesian account in which generalization is governed not by the raw number of circuit parameters, but by the effective dimension of the Fisher geometry induced by the circuit. This quantity is inflated by entanglement, making entangling connectivity an independent axis of complexity.In controlled experiments that fix the number of trainable rotations and vary only entanglement, we find that circuits with larger Fisher effective dimension exhibit larger train-test gaps, while parameter count is a weak predictor. The resulting bound acts primarily as a ranking certificate: it correctly orders circuits with identical parameter count, which parameter-counting bounds cannot do. We validate this mechanism across supervised classification, quantum contextual bandits, and value-function generalization, where entangled circuits consistently generalize worse than non-entangled circuits of equal parameter count, with gaps shrinking as sample size increases.Our strongest evidence comes from low-variance decision models, including single-observable classifiers, value heads, and one-step policies. In end-to-end multi-step policy learning, entanglement effects remain statistically significant but high return variance leaves the full ordering only partially resolved. Partial-correlation analysis shows that Fisher effective dimension screens off entangling pattern, and controls for training accuracy, readout, and optimizer rule out major optimization confounders. The effect also persists on an IBM Heron quantum processor under real noise. Overall, our results reframe quantum policy design around an entanglement--generalization trade-off rather than expressivity alone.