Gaussian process (GP) is a powerful modeling method with applications in machine learning for various engineering and non-engineering fields. Despite numerous benefits of modeling using GPs, the computational complexity associated with GPs demanding immense resources make their practical usage highly challenging. In this article, we develop a quantum version of multi-output Gaussian Process (QGP) by implementing a well-known quantum algorithm called HHL, to perform the Kernel matrix inversion within the Gaussian Process. To reduce the large circuit depth of HHL a circuit optimization technique called Approximate Quantum Compiling (AQC) has been implemented. We further showcase the application of QGP for a real-world problem to estimate line parameters of an electrical grid. Using AQC, up to 13-qubit HHL circuit has been implemented for a 32x32 kernel matrix inversion on IBM Quantum hardware for demonstrating QGP based line parameter estimation experimentally. Finally, we compare its performance against noise-less quantum simulators and classical computation results.
Gaussian process (GP) is a powerful modeling method with applications in machine learning for various engineering and non-engineering fields. Despite numerous benefits of modeling using GPs, the computational complexity associated with GPs demanding immense resources make their practical usage highly challenging. In this article, we develop a quantum version of multi-output Gaussian Process (QGP) by implementing a well-known quantum algorithm called HHL, to perform the Kernel matrix inversion within the Gaussian Process. To reduce the large circuit depth of HHL a circuit optimization technique called Approximate Quantum Compiling (AQC) has been implemented. We further showcase the application of QGP for a real-world problem to estimate line parameters of an electrical grid. Using AQC, up to 13-qubit HHL circuit has been implemented for a 32x32 kernel matrix inversion on IBM Quantum hardware for demonstrating QGP based line parameter estimation experimentally. Finally, we compare its performance against noise-less quantum simulators and classical computation results.
A Gaussian process (GP) is a powerful modeling method with applications in machine learning for various engineering and non-engineering fields.Despite numerous benefits of modeling using GPs, the computational complexity associated with GPs demanding immense resources make their practical usage highly challenging.In this article, a proof-of-concept demonstration of quantum-assisted multi-output Gaussian Process regression is presented, where the kernel matrix inversion during the training phase of GP is replaced by the Harrow-Hassidim-Lloyd (HHL) algorithm, a well-known quantum algorithm for solving linear systems.The presented approach uses classical kernels, but employs quantum computing for the computationally expensive matrix inversion operations.To reduce the large circuit depth of HHL, matrix conditioning and a circuit optimization technique called Approximate Quantum Compiling (AQC) is implemented, which reduces the circuit depth from O(10 7 ) gates to approximately 250 gates.The quantum-assisted Gaussian process is then utilized for implementation of line parameter of electrical grids, a real-world problem adapted from our previous work.Using AQC, up to 13-qubit HHL circuit has been implemented for a 32×32 kernel matrix inversion on IBM Quantum hardware (IBM Auckland).The performance of quantum hardware implementation is compared against noiseless quantum simulators and classical computation results.Through this work the feasibility of implementing quantum-assisted GP on current Noisy Intermediate-Scale Quantum (NISQ) hardware is demonstrated and key bottlenecks for future improvements are identified.To the best of authors' knowledge, this is one of the largest experimental implementations involving HHL algorithm for power system applications on IBM hardware.
A Gaussian process (GP) is a powerful modeling method with applications in machine learning for various engineering and non-engineering fields. Despite numerous benefits of modeling using GPs, the computational complexity associated with GPs demanding immense resources make their practical usage highly challenging. In this article, a proof-of-concept demonstration of quantum-assisted multi-output Gaussian Process regression is presented, where the kernel matrix inversion during the training phase of GP is replaced by the Harrow-Hassidim-Lloyd (HHL) algorithm, a well-known quantum algorithm for solving linear systems. The presented approach uses classical kernels, but employs quantum computing for the computationally expensive matrix inversion operations. To reduce the large circuit depth of HHL, matrix conditioning and a circuit optimization technique called Approximate Quantum Compiling (AQC) is implemented, which reduces the circuit depth fromO(107) gates to approximately 250 gates. The quantum-assisted Gaussian process is then utilized for implementation of line parameter of electrical grids, a real-world problem adapted from our previous work. Using AQC, up to 13-qubit HHL circuit has been implemented for a 32×32 kernel matrix inversion on IBM Quantum hardware (IBM Auckland). The performance of quantum hardware implementation is compared against noiseless quantum simulators and classical computation results. Through this work the feasibility of implementing quantum-assisted GP on current Noisy Intermediate-Scale Quantum (NISQ) hardware is demonstrated and key bottlenecks for future improvements are identified. To the best of authors’ knowledge, this is one of the largest experimental implementations involving HHL algorithm for power system applications on IBM hardware.
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.
Gaussian process regression (GPR) is a nonparametric supervised learning method, widely valued for its ability to quantify uncertainty. Despite its advantages and broad applications, classical GPR implementations face significant scalability challenges, as they involve matrix operations with a cubic complexity in relation to the dataset size. This computational challenge is further compounded by the demand of optimizing the Gaussian Process (GP) model over its hyperparameters, rendering the total computational cost prohibitive for data intensive problems. To address this issue, we propose a quantum gradient descent algorithm to optimize the GP model. Taking advantage of recent advances in quantum algorithms for linear algebra, our algorithm achieves exponential speedup in computing the gradients of the log marginal likelihood. The entire gradient descent process is integrated into the quantum circuit. Through runtime analysis and error bounds, we demonstrate that our approach significantly improves the scalability of GPR model optimization, making it computationally feasible for large-scale applications.
Multi-output time-series forecasting in energy systems is challenging because of nonlinear dynamics, multi-scale seasonality, and strong dependencies across correlated series. In this work, we investigate two hybrid quantum-classical frameworks for multi-stream time-series forecasting on a real Smart Meter dataset comprising 103 household electricity consumption time-series, with experiments executed on the ibm_marrakesh superconducting quantum processor. The first model, Kernelized Quantum Reservoir Computing with Repeated Measurement (KQRC-RM), combines coupled quantum reservoirs, ancilla-assisted repeated measurement, and kernelized readouts to model temporal dynamics and cross-stream correlations jointly. For a 3-stream time-series input and output, the KQRC-RM model using 114 qubits achieves an MAE of 0.0811 on MPS simulator (36.92\% improvement over its classical analog) whereas performance degrades to an MAE of 0.1524 on hardware. The second, a Projected Quantum Kernel Gaussian Process (QGP), replaces fidelity-based kernels with projected kernels constructed from local reduced-state statistics. Using a topology-aware 100-qubit QGP model to predict 100 multi-output time-series values, we observe 49\% of time-series outputs achieve high-accuracy predictions (MAE <0.15), with an average MAE of 0.082 for this low-error group. The medium-error regime (MAE 0.15-0.35) has an average MAE of 0.229, while the high-error regime (MAE >0.35) has an average MAE of 0.664. Overall, this reduces the average MAE relative to the classical GP baseline by 62.01\% on MPS simulator and 40.37\% on hardware. Together, these results demonstrate the feasibility of hybrid quantum machine learning for multi-input, multi-output time-series forecasting at the 100+ qubit scale on NISQ devices.
Gaussian Processes (GPs) are a powerful tool for probabilistic modeling, but their performance is often constrained in complex, large-scale real-world domains due to the limited expressivity of classical kernels. Quantum computing offers the potential to overcome this limitation by embedding data into exponentially large Hilbert spaces, capturing complex correlations that remain inaccessible to classical computing approaches. In this paper, we propose a Distributed Quantum Gaussian Process (DQGP) method in a multi-agent setting to enhance modeling capabilities and scalability. To address the challenging non-Euclidean optimization problem, we develop a Distributed consensus Riemannian Alternating Direction Method of Multipliers (DR-ADMM) algorithm that aggregates local agent models into a global model. We evaluate the efficacy of our method through numerical experiments conducted on a quantum simulator in classical hardware. We use real-world, non-stationary elevation datasets of NASA's Shuttle Radar Topography Mission and synthetic datasets generated by Quantum Gaussian Processes. Beyond modeling advantages, our framework highlights potential computational speedups that quantum hardware may provide, particularly in Gaussian processes and distributed optimization.