A Hybrid Quantum Solver for Gaussian Process Regression

TL;DR

This work tackles the cubic-time bottleneck of Gaussian process regression by introducing a hybrid quantum-classical approach, VQLS-GP, which uses the Variational Quantum Linear Solver to solve the linear systems arising in GP conditioning. It develops both global and local cost function formulations for VQLS, analyzes cost-evaluation costs, and demonstrates how state preparation, ansatz choices, and a kernel taper (MT kernel) reduce the quantum resource load. Empirical results on a one-dimensional Snelson benchmark show that VQLS-GP with the MT kernel and MUHEA ansatz can achieve regression performance close to classical GP MT while significantly lowering the required Pauli-string count and improving convergence relative to other configurations. The findings highlight the potential of near-term quantum devices to provide competitive probabilistic regression capabilities, and outline practical directions for scaling, hardware validation, and integration with quantum kernels.

Abstract

Gaussian processes are widely known for their ability to provide probabilistic predictions in supervised machine learning models. Their non-parametric nature and flexibility make them particularly effective for regression tasks. However, training a Gaussian process model using standard methods requires matrix inversions with a cubic time complexity, which poses significant computational challenges for inference on larger datasets. Quantum algorithms, such as the HHL algorithm, have been proposed as solutions that overcome the need for classical matrix inversions by efficiently solving linear systems of equations using quantum computers. However, to gain a computational advantage over classical algorithms, these algorithms require fault-tolerant quantum computers with a large number of qubits, which are not yet available. The Variational Quantum Linear Solver is a hybrid quantum-classical algorithm that solves linear systems of equations by optimizing the parameters of a variational quantum circuit using a classical computer. This method is especially suitable for noisy intermediate-scale quantum computers, as it does not require many qubits. It can be used to compute the posterior distribution of a Gaussian process by reformulating the matrix inversion into a set of linear systems of equations. We empirically demonstrate that using the Variational Quantum Linear Solver to perform inference for Gaussian process regression delivers regression quality comparable to that of classical methods.

Figures (8)

• Figure 1: HEA for a four-qubit example. This ansatz consists of $n(p+1)$ trainable parameters $\boldsymbol{\theta}$ with $n$ qubits and $p$ layers
• Figure 2: HEA with a single data reuploading (UHEA) for a four-qubit example. The unitary $U_{\mathbf{y}}$ embeds the training labels $\mathbf{y}$ and is applied before the HEA. The ansatz consists of $n(p+1)$ trainable parameters $\boldsymbol{\theta}$, with $n$ qubits and $p$ layers
• Figure 3: HEA with multiple data reuploadings (MUHEA) for a four-qubit example. The unitary $U_{\mathbf{y}}$ embeds the training labels $\mathbf{y}$ and is applied at the beginning of each layer. The ansatz consists of $n(p+1)$ trainable parameters $\boldsymbol{\theta}$, with $n$ qubits and $p$ layers
• Figure 4: GPR results for two GP and two VQLS-GP models applied to the Snelson test function (Eq. \ref{['eq:snelson']}), comparing the regression quality of the RBF (Eq. \ref{['eq:kernel_rbf']}) and MT (Eq. \ref{['eq:kernel_MT']}) kernels. The plots display the training samples (black dots), the predictive mean (colored solid line), the predictive uncertainty (shaded region), and the latent function (solid black line). (a) shows the regression result of a classical GP model with an RBF kernel. (b) depicts the result of our VQLS-GP model with an RBF kernel. (c) shows a classic GPR with an MT kernel. (d) shows the result of our VQLS-GP model with an MT kernel. The hyperparameters are optimized within the classical GPRs and reused for the VQLS-GP models without any further modifications. Both VQLS-GP models use the HEA shown in Fig. \ref{['fig:hea']}
• Figure 5: Averaged log-transformed loss curves for the VQLS-GP models of the kernel comparison in Section \ref{['sec5_kernel_comp']}. Each VQLS-GP run consists of $N=16$ VQLS optimization loops and each run was repeated $10$ times, resulting in a total of $160$ VQLS optimization loops for both the RBF and MT kernels. The corresponding regression results are depicted in Fig. \ref{['fig:gpr_kernels']} (b) and (d)