Quantum Reservoir Computing and Risk Bounds

TL;DR

This paper introduces bounds on the generalisation errors of quantum reservoirs using Rademacher complexity, analyzing how these bounds scale with the number of qubits and providing insights for controlling error in quantum machine learning models.

Contribution

It presents the first explicit, parameter-dependent generalisation bounds for quantum reservoirs, including analysis of their scaling with qubit number and applicability to various reservoir classes.

Findings

Risk bounds converge with increasing training samples for polynomial readout functions.

Bounds scale exponentially with the number of qubits, indicating challenges for large quantum systems.

Explicit parameter dependence allows for potential control of generalisation error.

Abstract

We propose a way to bound the generalisation errors of several classes of quantum reservoirs using the Rademacher complexity. We give specific, parameter-dependent bounds for two particular quantum reservoir classes. We analyse how the generalisation bounds scale with growing numbers of qubits. Applying our results to classes with polynomial readout functions, we find that the risk bounds converge in the number of training samples. The explicit dependence on the quantum reservoir and readout parameters in our bounds can be used to control the generalisation error to a certain extent. It should be noted that the bounds scale exponentially with the number of qubits $n$. The upper bounds on the Rademacher complexity can be applied to other reservoir classes that fulfill a few hypotheses on the quantum dynamics and the readout function.