Extending echo state property for quantum reservoir computing.

The echo state property (ESP) represents a fundamental concept in the reservoir computing (RC) framework that ensures output-only training of reservoir networks by being agnostic to the initial states and far past inputs. However, the traditional definition of ESP does not describe possible nonstationary systems in which statistical properties evolve. To address this issue, we present two categories of ESP: nonstationary ESP, designed for potentially nonstationary systems, and subspace and subset ESP, designed for systems whose subsystems have ESP. Following the definitions, we numerically demonstrate the correspondence between nonstationary ESP in the quantum reservoir computer (QRC) framework with typical Hamiltonian dynamics and input encoding methods using nonlinear autoregressive moving-average tasks. We also confirm the correspondence by computing linear or nonlinear memory capacities that quantify input-dependent components within reservoir states. Our study presents an understanding of the practical design of QRC and other possibly nonstationary RC systems in which nonstationary systems and subsystems are exploited.