We investigate a quantum reservoir computing (QRC) architecture based on a fully connected transverse-field Ising model for time series forecasting. Preliminary results on univariate financial data demonstrate that QRCenhanced models achieve improved predictive accuracy compared to a classical linear baseline. We outline planned extensions to multivariable time series forecasting using multi-qubit and compact encoding strategies, and propose future evaluations on both synthetic and real-world datasets.
Reservoir computing (RC) is a neural-network paradigm in which a fixed, high-dimensional dynamical
system, the reservoir, projects time-series inputs into a nonlinear feature space, while learning is
restricted to a linear readout layer [
While QRC has demonstrated strong performance in classification and univariate forecasting tasks [
In this work, we present preliminary results using QRC-enhanced models for univariate time series forecasting, demonstrating improved predictive accuracy over a classical linear baseline. These early ifndings motivate the development of QRC architectures for multivariable scenarios, where quantum reservoirs may provide eficient and scalable mechanisms for modeling inter-channel dependencies. We outline our planned extensions, including alternative encoding schemes, architectural variations, and evaluations on both synthetic and real-world multivariate datasets.
We implement a quantum reservoir (QR) based on a fully-connected transverse-efild Ising model,
following the protocol in [
In contrast to [
At each forecasting step, a normalized scalar input is sequentially encoded into the quantum reservoir by preparing the first qubit ( = 0) in the pure state | ()⟩ = √1 − |0⟩ + √ |1⟩, via amplitude encoding. The rest of the system remains unchanged. This updated local state is then tensored with the reduced density matrix of the remaining system, ↦→ | ()⟩ ⟨ ()| ⊗ Tr0[ ], efectively overwriting only the first qubit at each step.
The system subsequently evolves under the time-independent Hamiltonian = ∑︁ , + ∑︁ ℎ, < (1) where and are Pauli operators acting on qubit . The coeficients , represent inter-qubit couplings, and ℎ correspond to local transverse magnetic fields. In our implementation, the local ifelds are sampled from a uniform distribution [0.1, 1.0], while all inter-qubit couplings are fixed to , = 0.5 for ̸= , with , = 0.
This homogeneous configuration simplifies the reservoir architecture while preserving the fully
connected topology required for rich dynamics. Although [
The quantum state evolves over equally spaced time intervals (virtual nodes), during which the observables ⟨()⟩ = Tr[ ()] are extracted across all qubits. The resulting set of measurements forms a high-dimensional feature vector for each input. After processing a full input window, the linear readout weights are trained using the Moore–Penrose pseudoinverse solution to minimize the mean square error between predictions and targets.
We evaluated the performance of the quantum reservoir model on the S&P 500 aggregated daily closing
values time series [
To construct the input for the linear readout, we aggregated the reservoir features corresponding to the last four input injections, (− 3, − 2, − 1, ). The resulting high-dimensional feature vector was then used to predict the next 25 values of the time series, [+1, . . . , +25]. We evaluated the model for each time index ∈ [1100, 1199], using all data up to for training and the subsequent 25 points for testing.
The mean absolute error (MAE) was computed over the full prediction window for each value of ,
and the final score was obtained by averaging the MAE across all 100 indices. This averaging mitigates
the sensitivity of the MAE to the particular choice of , which can significantly afect forecasting
performance in financial time series [
To assess the added value of the quantum reservoir, we compared the full QRC pipeline to a baseline model where a linear readout was trained directly on the raw input values, without reservoir transformation. Each experiment was repeated across 200 independent quantum reservoir initializations to ensure statistical robustness. The average MAE results, shown in Fig. 1, demonstrate a consistent improvement of the QRC-enhanced model over the baseline, particularly at longer prediction horizons.
We presented a quantum reservoir computing (QRC) model based on a fully-connected transverse-field Ising Hamiltonian for univariate time series forecasting. Our implementation introduced a simplified and coherent reservoir design, with fixed inter-qubit couplings and product-state initialization in |+⟩⊗ , which led to improved predictive accuracy and stability across runs. Experiments on financial time series data show that QRC-enhanced models consistently outperform a baseline linear predictor trained directly on raw inputs. As shown in Fig. 1, the quantum reservoir significantly reduces the mean absolute error (MAE) for longer forecast steps, while maintaining comparable performance to the baseline in the short term. The results also demonstrate robustness across 200 randomly initialized reservoirs, as reflected by the confidence region.
Looking ahead, we aim to extend this framework to multivariable time series, which better reflect the complexity of real-world forecasting tasks. Possible architectural adaptations include multi-qubit input encoding, where each input channel is mapped to a separate qubit, and more compact encodings that embed multiple classical features into a single qubit, for example, via independent parameterized rotations. In addition, we will investigate how varying the reservoir size, particularly the number of qubits, influences predictive accuracy and dynamical richness.
We also plan to explore alternative physical realizations of the quantum reservoir, such as quantum
systems based on neutral atoms [
Finally, a promising future direction is to refine the prediction objective itself. Inspired by reward discounting in reinforcement learning, one could investigate loss functions that assign greater weight to near-term predictions within the forecasting horizon. This could potentially bias the model toward more stable or actionable outputs, depending on application needs. This work has been supported by grant PID2023-146520OB-C{1, 2}, funded by MICIU/AEI/10.13039/ 501100011033; by the Ministry for Digital Transformation and Civil Service of the Spanish Government through the QUANTUM ENIA project call – Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan – NextGenerationEU within the framework of the Digital Spain 2026 Agenda; and by ARQUIMEA Research Center and Horizon Europe, Teaming for Excellence, under grant agreement No 101059999, project QCircle. The work of Jesús Bonilla was partially supported by the Spanish Ministry of Science, Innovation and Universities through the Torres Quevedo grant PTQ2023-013228. The work of Jorge Ballesteros is financially supported by Instituto Tecnológico y de Energías Renovables (ITER) and Cabildo de Tenerife.
The authors have not employed any Generative AI tools.