Coherence influx is indispensable for quantum reservoir computing

Echo state property (ESP) is a fundamental concept for an input-driven dynamical system to perform information processing tasks. Recently, the authors proposed extensions of ESP to possibly non-stationary systems and subsystems. In this paper, as a follow-up, we theoretically analyze sufficient and necessary conditions for a quantum system to satisfy non-stationary ESP and subset/subspace non-stationary 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 to realize the non-stationary ESP, and 2) the spectral radius of PTM can characterize the fading memory property of 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 RC (mRC), an RC system with one-dimensional multiplicative input. Theoretically and numerically, we show that the corresponding parameter to spectral radius/coherence influx in mRC directly correlates with its linear memory capacity (MC). Our findings will enlighten a theoretical aspect of PTM and the multiplicative nature of QRC. They will lead to a further understanding of QRC and information processing in open quantum systems.