Reversible Information Transformation via Quantum Reservoir Computing: Conditions, Protocol, and Noise Resilience

Abstract: Quantum reservoir computing (QRC) exploits fixed quantum dynamics and a trainable linear readout to process temporal data, yet reversing the transformation -- reconstructing the input from the reservoir output -- has been considered intractable owing to the recursive nonlinearity of sequential quantum state evolution. Here we propose a four-equation encode-decode protocol with cross-key pairing and constructively show that quantum reservoir and key combinations satisfying all four equations exist. Using a full XYZ Hamiltonian reservoir with 10 data qubits, we expand the feature dimension to 76 without increasing qubit count and achieve machine-precision reconstruction (mean-squared error $\mathrm{MSE} \sim 10^{-17}$) for data lengths up to 30 under ideal conditions; the rank condition $\mathrm{dim}(V) \geq N_c$ is identified as a necessary criterion. A comprehensive noise analysis across seven conditions and four baseline methods reveals a clear hierarchy: shot noise dominates, depolarizing noise adds a moderate factor, and asymmetric resource allocation -- 10 shots for encoding, $10^5$ for decoding -- yields approximately two orders of magnitude MSE improvement by exploiting the asymmetric noise roles of the encryption and decryption feature matrices. Under realistic noise the MSE degrades to $10^{-3}$-$10^{-1}$, indicating that error mitigation is needed before practical deployment, but our results establish the feasibility of bidirectional reversible information transformation within QRC.