Expressivity Limits of Quantum Reservoir Computing

Abstract: We investigate the fundamental expressivity limits of quantum reservoir computing (QRC) by establishing a formal connection to parametrized quantum circuit quantum machine learning (PQC-QML). We analytically prove, and numerically corroborate, that in QRC the number of orthogonal non-linear functions that can be generated from classical data is bounded linearly by the number of input encoding gates, independent of the reservoir's Hilbert space size. This finding applies across both physical and gate-based reservoir implementations using typical single-qubit input rotation schemes. Our results challenge the common assumption that exponential Hilbert space scaling confers a corresponding computational advantage in QRC, and demonstrate that true quantum benefit will require either more sophisticated, potentially multi-qubit, input schemes or quantum-native input data. These insights lay new groundwork for the design and evaluation of future QRC hardware and algorithms.