Quantum reservoir computing in finite dimensions

TL;DR

This paper explores alternative mathematical representations for quantum reservoir computing, establishing isomorphisms that connect density matrix and observable space approaches, leading to new insights into system properties like fading memory and echo state conditions.

Contribution

It introduces vector representations that unify classical and quantum reservoir computing theories, providing necessary and sufficient conditions for key properties in finite-dimensional systems.

Findings

Vector representations yield state-affine systems in quantum reservoir computing.

Fading memory and echo state properties are representation-independent.

Characterization of contractive quantum channels with trivial solutions.

Abstract

Most existing results in the analysis of quantum reservoir computing (QRC) systems with classical inputs have been obtained using the density matrix formalism. This paper shows that alternative representations can provide better insights when dealing with design and assessment questions. More explicitly, system isomorphisms are established that unify the density matrix approach to QRC with the representation in the space of observables using Bloch vectors associated with Gell-Mann bases. It is shown that these vector representations yield state-affine systems (SAS) previously introduced in the classical reservoir computing literature and for which numerous theoretical results have been established. This connection is used to show that various statements in relation to the fading memory (FMP) and the echo state (ESP) properties are independent of the representation, and also to shed some light on fundamental questions in QRC theory in finite dimensions. In particular, a necessary and sufficient condition for the ESP and FMP to hold is formulated using standard hypotheses, and contractive quantum channels that have exclusively trivial semi-infinite solutions are characterized in terms of the existence of input-independent fixed points.