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.
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.
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.
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 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.
We propose a hybrid quantum-classical implementation of the Harrow-Hassidim-Lloyd (HHL) algorithm for the matrix inversion (linear-systems) problem, a key subroutine in quantum computing. Rather than claiming a new complexity result, we provide an assumption-explicit proof sketch, aligned with established literature, that contextualizes matrix inversion as Bounded-Error Quantum Polynomial Time (BQP)-complete and offers reduction-based intuition via Grover-type amplitude amplification. We implement HHL for 2 × 2, 4 × 4, and 8 × 8 Hermitian matrices using Qiskit and evaluate performance using simulator-based experiments with calibrated noise models and executions on IBM Quantum and IonQ backends. To enhance robustness on Noisy Intermediate-Scale Quantum (NISQ) hardware, we integrate measurement-error mitigation, zero-noise extrapolation, and post-selection, and we quantify performance using fidelity and mean squared error. Results from noiseless simulation, calibrated-noise simulation, and real-device execution show that mitigation improves reconstruction fidelity for small Hermitian systems, particularly for deeper circuits. These findings support mitigation-aware feasibility on current NISQ devices; however, they do not establish present-day runtime advantage over optimized classical solvers. The proposed workflow is relevant to quantum optimization, quantum machine learning, and quantum chemistry, where linear-system primitives arise frequently. Future work will focus on reducing phase-estimation depth (e.g., iterative or semiclassical variants), improving compilation and benchmarking transparency, and re-evaluating larger instances on higher-fidelity hardware.
The rapid integration of renewable energy resources presents formidable challenges in managing power grids. While advanced computing and machine learning techniques offer some solutions for accelerating grid modeling and simulation, there remain complex problems that classical computers cannot effectively address. Quantum computing, a promising technology, has the potential to fundamentally transform how we manage power systems, especially in scenarios with a higher proportion of renewable energy sources. One critical aspect is solving large-scale linear systems of equations, crucial for power system applications like power flow analysis, for which the Harrow-Hassidim-Lloyd (HHL) algorithm is a well-known quantum solution. However, HHL quantum circuits often exhibit excessive depth, making them impractical for current Noisy-Intermediate-Scale-Quantum (NISQ) devices. In this paper, we introduce a versatile framework, powered by NWQSim, that bridges the gap between power system applications and quantum linear solvers available in Qiskit. This framework empowers researchers to efficiently explore power system applications using quantum linear solvers. Through innovative gate fusion strategies, reduced circuit depth, and GPU acceleration, our simulator significantly enhances resource efficiency. Power flow case studies have demonstrated up to a eight-fold speedup compared to Qiskit Aer, all while maintaining comparable levels of accuracy.
Gaussian processes (GPs) are a widely used model for regression problems in supervised machine learning. Implementation of GP regression typically requires O(n3) logic gates. We show that the quantum linear systems algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] can be applied to Gaussian process regression (GPR), leading to an exponential reduction in computation time in some instances. We show that even in some cases not ideally suited to the quantum linear systems algorithm, a polynomial increase in efficiency still occurs.