Continuous-Time Quantum Reservoirs

This paper studies temporal learning with quantum reservoirs driven by continuous-time dynamics. Our goal is to model long-range dependencies under irregular sampling while avoiding backpropagation through time and repeated numerical ODE integration. We formulate a Continuous-time Quantum Reservoir (CQR) framework in which an input-modulated Hamiltonian drives Schrödinger evolution, and a classical linear readout is fitted from sampled quantum features. We analyze three variants: (i) CQR-unitary, which uses expectation values of Hermitian observables; (ii) CQR-measurement, which uses projective-measurement probabilities; and (iii) CQR-trainable, which additionally tunes input-coupling Hamiltonian parameters via parameter-shift updates. Experiments on Lorenz forecasting, human-activity classification, and two regression benchmarks show competitive accuracy, improved memory capacity, and clear compute-accuracy trade-offs across variants. We also provide a Stone-Weierstrass-based universality result for the induced feature algebra and discuss practical design limitations.