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

TL;DR

A four-equation encode-decode protocol with cross-key pairing is proposed and constructively show that quantum reservoir and key combinations satisfying all four equations exist, establishing the feasibility of bidirectional reversible information transformation within QRC.

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.

Figures (13)

• Figure 1: Schematic of the reversible QRC protocol. Path 1 encrypts with key $A$ on reservoir $a$ and decrypts with key $\beta$ on reservoir $b$. Path 2 encrypts with key $B$ on reservoir $b$ and decrypts with key $\alpha$ on reservoir $a$. The cross structure (dashed arrows) couples the two paths through the intermediate ciphertexts $\gamma$ and $\gamma'$. The four equations [Eqs. \ref{['eq:enc1']}--\ref{['eq:dec1']}] must be simultaneously satisfied for successful reconstruction.
• Figure 2: Pseudocode for the iterative solving algorithm. The encryption feature matrices (steps 2--3) are computed once; the decryption feature matrices (steps 7, 11) are recomputed at each iteration as the intermediate ciphertexts $\gamma$, $\gamma'$ are updated.
• Figure 3: Quantum circuit architectures (representative qubits shown; full circuits use 11 qubits). Each circuit is repeated for every timestep $t = 1, \ldots, N_c$. (a) Hénon map / delay-time embedding + quantum circuit: $[R_Y{\to}\mathrm{CNOT\;ladder}{\to}R_Z]{\times}3$ (66 params); classical preprocessing before $R_Z$ encoding on the ancilla. (b) Tree tensor network (TTN): binary tree of 10 two-qubit unitary blocks (240 variational params). (c) QRC XYZ Hamiltonian reservoir: time evolution $e^{-iH\Delta t}$ with 1--4 body Pauli interactions; two circuits alternate with period 6 (2,888 fixed params); $d{=}76$ features including $\langle Z_iZ_j\rangle$ correlators; quantum state carries over between timesteps.
• Figure 4: Ideal condition (Exp 1) reconstruction MSE as a function of data length $N_c$ (Path 1). Machine-precision reconstruction ($\mathrm{MSE} \sim 10^{-17}$--$10^{-18}$) is achieved for all $N_c \leq 30$. The sharp degradation at $N_c = 35$ reflects the effective rank limitation of the feature matrix. Results are averaged over 16 random Hamiltonian seeds $\times$ 3 trials; error bars indicate one standard deviation. Path 2 results (not shown) are quantitatively equivalent.
• Figure 5: Shot noise condition (Exp 2, Path 1): $N_{\mathrm{shots}} = 1{,}000$. The transition from ideal (Fig. \ref{['fig:ideal']}) to shot noise introduces a degradation of $\sim 10^{16}$ orders of magnitude. MSE values of $\sim 10^{-1}$ are consistent with the shot noise variance $\sigma^2 \propto 1/N_{\mathrm{shots}}$ propagated through the $76 \times N_c$ feature matrix elements.