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: non-stationary ESP designed for possibly non-stationary systems, and subspace/subset ESP designed for systems whose subsystems possess ESP. In the quantum reservoir computing (QRC) framework under typical Hamiltonian dynamics and input encoding methods, we numerically demonstrate the correspondence between non-stationary ESP and the performance of nonlinear autoregressive moving average (NARMA) and linear/nonlinear memory tasks. Our findings reveal that the tendencies observed in the root normalized mean squared error (RNMSE) of the NARMA task closely aligned with the distribution of the non-stationary ESP indicator. Furthermore, metrics such as total memory capacity (MC) and total information processing capacity (IPC), derived from linear and nonlinear memory tasks, exhibit behaviors consistent with non-stationary ESP with partial exceptions. It suggests that the non-stationary ESP explains at least a large part of the system's information processing capability. Our proposed categories of ESP provide a new understanding of the practical design of QRC, and other RC systems that may exhibit some type of non-stationarity.