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

2026-07-07

AI Summary

This work addresses the unclear mechanisms underlying generalization in quantum reinforcement learning with parameterized quantum circuits. For the first time, entanglement is identified as a structural complexity axis influencing generalization. Building upon the PAC-Bayesian framework and Fisher information geometry, the study proposes a novel theoretical perspective in which the Fisher effective dimension—not merely the number of parameters—governs generalization performance, yielding generalization bounds capable of distinguishing circuits with identical parameter counts but differing structures. Controlled experiments and partial correlation analyses on IBM’s Heron quantum processor empirically validate that increased entanglement elevates the Fisher effective dimension, leading to degraded generalization and a wider train-test gap even when parameter counts are held constant. This detrimental effect remains pronounced in low-variance models and persists under realistic hardware conditions.

Abstract

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.

Problem

• quantum reinforcement learning
• generalization
• entanglement
• parameterized quantum circuits
• PAC-Bayesian

Innovation

• PAC-Bayesian
• Fisher effective dimension
• entanglement
• quantum reinforcement learning
• generalization bound