Coherence influx is indispensable for quantum reservoir computing

TL;DR

This paper investigates the conditions under which quantum reservoir computing exhibits the echo state property, emphasizing the role of coherence influx and spectral radius, and introduces a simplified model to analyze these mechanisms.

Contribution

It provides a theoretical and numerical analysis of nonstationary ESP in quantum systems, highlighting the importance of coherence influx and spectral radius, and proposes a simplified multiplicative reservoir model.

Findings

Coherence influx is essential for nonstationary ESP in quantum systems.

Spectral radius of PTM characterizes the fading memory and phase transitions.

The simplified mRC model links spectral radius and coherence influx to memory capacity.

Abstract

Echo state property (ESP) is a fundamental property that allows an input-driven dynamical system to perform information processing tasks. Recently, extensions of ESP to potentially nonstationary systems and subsystems, that is, nonstationary ESP and subset/subspace ESP, have been proposed. In this paper, we theoretically and numerically analyze the sufficient and necessary conditions for a quantum system to satisfy nonstationary ESP and subset/subspace nonstationary ESP. Based on extensive usage of the Pauli transfer matrix (PTM) form, we find that (1) the interaction with a quantum-coherent environment, termed $\textit{coherence influx}$, is indispensable in realizing nonstationary ESP, and (2) the spectral radius of PTM can characterize the fading memory property of quantum reservoir computing (QRC). Our numerical experiment, involving a system with a Hamiltonian that entails a spin-glass/many-body localization phase, reveals that the spectral radius of PTM can describe the dynamical phase transition intrinsic to such a system. To comprehensively understand the mechanisms under ESP of QRC, we propose a simplified model, multiplicative reservoir computing (mRC), which is a reservoir computing (RC) system with a one-dimensional multiplicative input. Theoretically and numerically, we show that the parameters corresponding to the spectral radius and coherence influx in mRC directly correlates with its linear memory capacity (MC). Our findings about QRC and mRC will provide a theoretical aspect of PTM and the input multiplicativity of QRC. The results will lead to a better understanding of QRC and information processing in open quantum systems.