Quantum State-Space Machines: A Channel-Based Framework for Learning, Memory, and Quantum Advantage

We study quantum state-space models (QSSMs) as recurrent architectures for sequence pro cessing, with an emphasis on the interplay between quantum dynamics, measurement design, and readout expressivity. We consider a delayed XOR benchmark, a canonical task for probing long-range memory and parity-based dependencies, and implement a noiseless, statevector-based quantum reservoir driven by input-conditioned unitary evolution. Using expectation values of local Pauli observables as features, we find that quantum state space dynamics alone remain near chance level on the delayed XOR task, despite the high dimensional and nonlinear nature of the underlying quantum evolution. This result highlights a fundamental observability limitation: parity information is encoded in higher-order correlations that are not linearly accessible through simple expectation-value measurements. We then demonstrate that augmenting the quantum features with a minimal tapped-delay readout, including a bilinear interaction term between delayed inputs, restores perfect gener alization. This augmentation does not modify the quantum dynamics, but instead aligns the observation model with the algebraic structure of the task. Our findings clarify the respective roles of quantum dynamics and measurement interfaces in quantum recurrent models, and suggest that performance gains arise not solely from quantum evolution, but from the coherent integration of dynamics, observability, and readout design.