QuaCK-TSF: Quantum-Classical Kernelized Time Series Forecasting

TL;DR

QuaCK-TSF tackles probabilistic time-series forecasting by marrying Gaussian process regression with a quantum kernel built from an IQP-inspired feature map that encodes temporal sub-windows. Hyperparameters are efficiently tuned using gradient-free Bayesian optimization, leveraging a Matérn-5/2 GP surrogate and a Sobol-based space-filling design. Empirical results on synthetic data show competitive performance against classical kernels and the highest log-likelihood, confirming the quantum kernel's capacity to capture nonlinear temporal dynamics and provide uncertainty quantification. The study demonstrates a practical route to quantum-enhanced probabilistic TSF on NISQ-like hardware and underscores the importance of kernel design and optimization in realizing quantum advantages for time-series tasks.

Abstract

Forecasting in probabilistic time series is a complex endeavor that extends beyond predicting future values to also quantifying the uncertainty inherent in these predictions. Gaussian process regression stands out as a Bayesian machine learning technique adept at addressing this multifaceted challenge. This paper introduces a novel approach that blends the robustness of this Bayesian technique with the nuanced insights provided by the kernel perspective on quantum models, aimed at advancing quantum kernelized probabilistic forecasting. We incorporate a quantum feature map inspired by Ising interactions and demonstrate its effectiveness in capturing the temporal dependencies critical for precise forecasting. The optimization of our model's hyperparameters circumvents the need for computationally intensive gradient descent by employing gradient-free Bayesian optimization. Comparative benchmarks against established classical kernel models are provided, affirming that our quantum-enhanced approach achieves competitive performance.

Figures (9)

• Figure 1: The IQP feature map $\mathcal{U}(\mathbf{x}, \alpha)$ constructs the quantum state $\ket{\phi(\mathbf{x}, \alpha)}$ by sequentially applying two layers of operations across all qubits: first, Hadamard gates are applied simultaneously to each qubit, followed by the collective application of the unitary $U_z(\mathbf{x}, \alpha)$.
• Figure 2: Quantum kernel realization via fidelity state overlap, wherein the circuit evaluates fidelity by sequentially applying unitary $\mathcal{U}(\mathbf{x}, \alpha)$, its inverse $\mathcal{U}^\dag(\mathbf{x}', \alpha)$, and estimating the probability of an all-zero measurement outcome.
• Figure 3: The synthetic time series, featuring multiple non-linear patterns, is divided into two sets: the training set $\mathcal{D}$, shown as the blue curve, and the testing set $\mathcal{D}'$, represented by the orange curve.
• Figure 4: The IQP-based GP's mean predictions are depicted by the dark blue curve, with the light shaded area indicating the predictions' 95% confidence interval.
• Figure 5: Bayesian optimization contour plots show evaluated configurations from two perspectives with a color gradient from white to dark green representing MLL values—darker shades indicate higher values, and square markers denote objective function evaluations.