Hierarchy of the echo state property in quantum reservoir computing

The echo state property (ESP) represents a fundamental concept in the reservoir computing framework that ensures stable output-only training of reservoir networks. However, the conventional definition of ESP does not aptly describe possibly non-stationary systems, where statistical properties evolve. To address this issue, we introduce two new categories of ESP: $\textit{non-stationary ESP}$ designed for possibly non-stationary systems, and $\textit{subspace/subset ESP}$ designed for systems whose subsystems have ESP. Following the definitions, we numerically demonstrate the correspondence between non-stationary ESP in the quantum reservoir computer (QRC) framework with typical Hamiltonian dynamics and input encoding methods using nonlinear autoregressive moving-average (NARMA) tasks. These newly defined properties present a new understanding toward the practical design of QRC and other possibly non-stationary RC systems.