Multivariate Time Series Forecasting with Adaptive Non-Local Observables

Quantum neural networks (QNNs) are typically built from variational quantum circuits (VQCs), which are limited by local measurements. Adaptive non-local observables (ANO) address this by jointly optimizing circuit parameters and multi-qubit measurements. However, existing ANO-based VQCs learn only a single static observable that remains invariant across all inputs. We propose QFWP-ANO, a novel architecture which employs a classical hypernetwork to dynamically program VQC parameters and/or non-local observables conditioned on each input. On multivariate time-series forecasting across four ETT datasets, QFWP-ANO achieves the lowest MSE in 16 of 20 settings and second-lowest in the remaining four, surpassing ANO-based and other strong baselines. On reinforcement learning tasks, QFWP-ANO consistently surpasses ANO-VQCs. Our results establish input-conditioned ANO as an effective approach for enhancing QNNs.

This paper presents a unified quantum-classical hybrid framework for multi-horizon time-series forecasting, introducing two model variants Quantum Reservoir Forecaster (QRC-F) and Variational Quantum Forecaster (VQF-F). The proposed framework investigates the complexity-fidelity trade-off of quantum forecasting under near-term NISQ hardware constraints. Continuous time-series signals are transformed into binary representations through uniform quantization and encoded into quantum states using angle encoding with parameterized RY rotation gates. Cross-channel entanglement layers capture dependencies among multiple variables. QRC-F utilizes a fixed random unitary quantum reservoir for stable, gradient-free temporal feature extraction, whereas VQF-F employs a trainable variational quantum circuit optimized through the parameter-shift rule to learn temporal and inter-variable patterns from Pauli expectation values. Both models replace computationally expensive quadratic self-attention with efficient linear transformations, reducing parameter complexity. A shared MIMO-based multi-horizon prediction head simultaneously generates forecasts across multiple horizons, avoiding error accumulation in recursive forecasting. Experimental evaluations on benchmark datasets, including ETTh1, ETTh2, ETTm1, ETTm2, Weather, electricity, and exchange-rate, demonstrate that VQF-F achieves superior training stability and parameter efficiency, while QRC-F provides enhanced robustness and circuit fidelity under quantum noise. The results establish a practical quantum-native forecasting framework with strong potential for deployment on near-term NISQ devices.

The analysis of Multivariate Time Series (MTS) plays an important role in a lot of real-world practical applications, but it still remains some challenging problem about capturing multi-granularity structural patterns and suppressing noise appropriately. Multi-Scale Convolution with Optimal Transport Attention (MSC-OT) is proposed in this paper. MSC-OT is a useful architecture to optimize the attention mechanism. It combines multi-scale convolution with Sinkhorn optimal transport method based on inverted embedding. The inverted embedding approach embeds each variable as a token and allows the model to capture cross-variate relationships better. MSC-OT consists of two part: (1) Multi-Scale Convolution Enhancement, that applies multi-scale convolutions to attention score matrices based on inverted embedding, capturing local structural patterns in the variate-interaction space induced by compressed temporal representations; (2) Sinkhorn Optimal Transport Regularization, that formulates attention computation as an optimal transport problem and employs iterative matrix scaling to ensure balanced information flow across variates. Adaptive Fusion Strategy utilizes softmax-normalized learnable weights to dynamically combine base attention, convolution-enhanced, and OT-regularized scores. Experiments on widely-used datasets, including ETT, Electricity, Traffic, Solar-Energy, and Exchange-Rate, show that MSC-OT achieves well performance in both short-term and long-term forecasting tasks. Ablation experiments further validate the effectiveness of each proposed component and their synergistic contributions to improving prediction accuracy for multivariate time series forecasting.