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 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.
Abstract Gaussian processes are widely used probabilistic models for regression and classification tasks. Their flexibility and ability to quantify uncertainties make them well-suited for many applications. However, their applicability to large datasets is limited by the cubic computational time complexity. While reduced-rank methods such as Hilbert space approximations reduce this complexity, scaling to very large datasets remains challenging. Quantum algorithms are promising candidates to further accelerate such computations. Quantum principal component analysis has previously been applied to Hilbert space Gaussian process regression to address this scaling limitation. However, its applicability is limited by the requirement for fault-tolerant quantum hardware. In this work, we present a hybrid quantum-classical model for reduced-rank Gaussian process regression, where we apply a quantum principal component analysis algorithm based on parameterized quantum circuits to Hilbert space Gaussian process regression. In noiseless simulations on a one-dimensional test function, the resulting model reaches a regression accuracy comparable to that of the exact Gaussian process and the classical Hilbert space Gaussian process. The framework uses shallow parameterized circuits and does not require fault-tolerant quantum hardware, which makes it a candidate for noisy intermediate-scale quantum devices.
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.
Abstract Variational quantum algorithms are promising methods with the greatest potential to achieve quantum advantage, widely employed in the era of noisy intermediate-scale quantum computing. This study presents an advanced variational hybrid algorithm (EVQLSE) that leverages both quantum and classical computing paradigms to address the solution of linear equation systems. Initially, an innovative loss function is proposed, drawing inspiration from the similarity measure between two quantum states. This function exhibits a substantial improvement in computational complexity when benchmarked against the variational quantum linear solver. Subsequently, a specialized parameterized quantum circuit structure is presented for small-scale linear systems, which exhibits powerful expressive capabilities. Through rigorous numerical analysis, the expressiveness of this circuit structure is quantitatively assessed using a variational quantum regression algorithm, and it obtained the best score compared to the others. Moreover, the expansion in system size is accompanied by an increase in the number of parameters, placing considerable strain on the training process for the algorithm. To address this challenge, an optimization strategy known as quantum parameter sharing is introduced, which proficiently minimizes parameter volume while adhering to exacting precision standards. Finally, EVQLSE is successfully implemented on a quantum computing platform provided by IBM for the resolution of large-scale problems characterized by a dimensionality of 220.
A hybrid quantum–classical approach to solving the load balancing problem in real-time computer systems is proposed, with a focus on reliability assessment and control. The approach is based on the use of a Variational Quantum Linear Solver (VQLS) to process large sparse linear systems arising from the spatio-temporal discretization of load distribution models and task flows among computational nodes. An algorithmic scheme is presented for transforming an engineering model of a real-time system into a quantum-realizable problem formulation. Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. This scheme includes decomposing the system matrix into a linear combination of Pauli operators, preparing a quantum state corresponding to the right-hand side, and optimizing a variational ansatz. It is shown that the resulting quantum solution state is most effective when used not for full reconstruction of the load vector, but for direct estimation of aggregated and local metrics, such as peak load, slack to deadline violations, and integral stability indicators, which are directly related to system failures and performance degradation. The proposed approach is considered as a tool for quantum benchmarking of classical load balancing algorithms, as well as a potential foundation for building reliability management systems for computer systems operating in a near–real-time regime. Key words. Reliability of computer systems, Bayesian neural networks, ensemble methods, artificial intelligence, hybrid models, risk assessment, fault tolerance, uncertainty modeling, failure prediction, software security, system resilience, neural networks, fault trees, machine learning, quantum systems, quantum algorithms, quantum machine learning, hybrid models.
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 (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.