quant-phApr 3, 2026

Recurrent Quantum Feature Maps for Reservoir Computing

Authors: Utkarsh Singh, Aaron Z. Goldberg, Christoph Simon, Khabat Heshami

Organizations: Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario, K1N 6N5 Canada · National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1N 5A2, Canada · Institute for Quantum Science and Technology, Department of Physics and Astronomy, University of Calgary, Alberta T2N 1N4, Canada · Hotchkiss Brain Institute, University of Calgary, Alberta T2N 1N4, Canada

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Abstract

Reservoir computing promises a fast method for handling large amounts of temporal data. This hinges on constructing a good reservoir--a dynamical system capable of transforming inputs into a high-dimensional representation while remembering properties of earlier data. In this work, we introduce a reservoir based on recurrent quantum feature maps where a fixed quantum circuit is reused to encode both current inputs and a classical feedback signal derived from previous outputs. We evaluate the model on the Mackey-Glass time-series prediction task using our recently introduced CP feature map, and find that it achieves lower mean squared error than standard classical baselines, including echo state networks and multilayer perceptrons, while maintaining compact circuit depth and qubit requirements. We further analyze memory capacity and show that the model effectively retains temporal information, consistent with its forecasting accuracy. Finally, we study the impact of realistic noise and find that performance is robust to several noise channels but remains sensitive to two-qubit gate errors, identifying a key limitation for near-term implementations.

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Figure 1: Schematic representation of the classical reservoir computing framework. The input vector xt\mathbf{x}_{t} is projected into a high-dimensional dynamical space by a recurrent network of fixed, randomly connected internal nodes (the reservoir), characterized by weights Win\mathbf{W}_{\mathrm{in}} and Wres\mathbf{W}_{\mathrm{res}} . The resulting reservoir states are then linearly mapped to the target outputs yj\mathbf{y}_{j} via a trainable readout layer with weights Wout\mathbf{W}_{\mathrm{out}} . Only the output layer is optimized during training, while the reservoir dynamics remain untrained, enabling efficient learning of complex temporal patterns.
Figure 2: Schematic of proposed feedback-driven quantum reservoir computing model. At each timestep tt , a windowed input sequence [xt−τ,…,xt][x_{t-\tau},\dots,x_{t}] is encoded into the left half of a quantum feature map circuit U(⋅)U(\cdot) , while the right half applies the inverted circuit U†(⋅)U^{\dagger}(\cdot) using a feedback vector derived from the previous reservoir output. The feedback is modulated by a scaling parameter α∈[0,1]\alpha\in[0,1] , with α=1\alpha=1 corresponding to full feedback and α=0\alpha=0 to input-only evolution. The quantum circuit is initialized in ∣0⟩⊗n|0\rangle^{\otimes n} , and measurement outcomes are collected to produce a classical output vector, which goes to the regression model to generate the prediction yty_{t} .
Figure 3: Model comparison on the Mackey–Glass dataset at τ=17\tau=17 , window size = 20, prediction horizon = 20. Quantum reservoir achieves the lowest MSE despite no hyperparameter tuning.
Figure 4: Predicted vs. true signal for the quantum reservoir over 100 test time steps. The reservoir captures the chaotic dynamics with high fidelity and stability.
Figure 5: Mean squared error (MSE) across Mackey–Glass delays τ=15\tau=15 to 5050 , for various prediction horizons, and two different feature maps. The CPMap consistently outperforms the ZZFeatureMap across delays and horizons.
Figure 6: Impact of quantum circuit parameter θi\theta_{i} on the reservoir’s entanglement structure, memory capacity, and predictive performance. A single parameter θi∈[−π/2,π/2]\theta_{i}\in[-\pi/2,\pi/2] is swept while keeping all other circuit parameters fixed. The average single-qubit entanglement entropy (dashed blue) is computed across all qubits, the memory capacity (green) is derived from the explicit STM formulation, and MSE (solid red) quantifies forecasting error. The results reveal a structured dependence of the reservoir’s computational behavior on θi\theta_{i} , with distinct parameter regions exhibiting enhanced memory retention, moderate entanglement, and low prediction error.
Figure 7: Predicted versus true Mackey-Glass signal for the quantum reservoir under noisy simulation at τ=17\tau=17 . The noisy prediction preserves the coarse oscillatory trend but exhibits strong local fluctuations and irregular spikes, indicating degradation of fine temporal structure under realistic device noise.
Figure 8: Performance (MSE score) of the quantum reservoir as a function of noise strength for different noise channels. Single-qubit and readout noise have little effect, whereas two-qubit depolarizing noise leads to a sharp degradation. The combined-noise case shows the strongest deterioration.
Figure 9: Worst-case MSE for each noise type at the maximum tested noise parameter. Two-qubit and combined noise dominate the degradation.
Figure 10: Variation of test MSE with readout limit λ\lambda (top) and feedback strength α\alpha (bottom). The best performance is achieved at (α,λ)=(0.79,1.0)(\alpha,\lambda)=(0.79,1.0) , demonstrating the importance of both recurrence and output dimensionality in quantum reservoir dynamics.
Figure 11: Dynamical analysis of the quantum reservoir. (a) Memory capacity increases with window size and saturates around 20, indicating a limit in temporal retention. (b) Convergence of reservoir trajectories confirms that the quantum system satisfies the echo state property.
Figure 12: MSE as a function of relaxation time T1T_{1} . The weak dependence on T1T_{1} indicates that relaxation noise has a limited impact on performance in the explored regime.
Figure 13: Model comparison on the Mackey–Glass dataset at τ=17\tau=17 , window size = 20, prediction horizon = 20. CPQRC-lite achieves lower error than classical reservoir computing and performs comparably to MLP, while the original CPQRC model attains the lowest MSE.
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