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arXiv:1512.03929 · Physical Review A

Quantum-assisted Gaussian process regression

Zhikuan Zhao, Jack Fitzsimons, Joseph F. Fitzsimons

Published May 22, 2019• 101 citations
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Abstract

Gaussian processes (GPs) are a widely used model for regression problems in supervised machine learning. Implementation of GP regression typically requires $O({n}^{3})$ 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.

DOI
10.1103/physreva.99.052331

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Gaussian processes (GPs) are a widely used model for regression problems in supervised machine learning. Implementation of GP regression typically requires O(n3)O({n}^{3})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.
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.
Gaussian processes (GPs) are important models in supervised machine learning. Training in Gaussian processes refers to selecting the covariance functions and the associated parameters in order to improve the outcome of predictions, the core of which amounts to evaluating the logarithm of the marginal likelihood (LML\mathrm{LML}LML) of a given model. The LML\mathrm{LML}LML gives a concrete measure of the quality of prediction that a GP model is expected to achieve. The classical computation of the LML\mathrm{LML}LML typically carries a polynomial time overhead with respect to the input size. We propose a quantum algorithm that computes the logarithm of the determinant of a Hermitian matrix, which runs in logarithmic time for sparse matrices. This is applied in conjunction with a variant of the quantum linear system algorithm that allows for logarithmic-time computation of the form ${\mathbf{y}}^{T}{A}^{\ensuremath{-}1}\mathbf{y}$, where y\mathbf{y}y is a dense vector and AAA is the covariance matrix. We hence show that quantum computing can be used to estimate the LML\mathrm{LML}LML of a GP with exponentially improved efficiency under certain conditions.
The Gaussian process is a widely used model for regression problems in supervised machine learning. However, predicting new inputs via a Gaussian process model becomes computationally inefficient when training a large data set. This paper proposes a fast quantum algorithm for prediction based on the Gaussian process regression. The proposed quantum algorithm consists of two subalgorithms: the first one aims to efficiently prepare the squared exponential covariance matrices and covariance functions vector with annihilation and creation operators; the other is to obtain predictive mean values and covariance values for new inputs. Evidence is also shown that the proposed quantum Gaussian process regression algorithm can achieve quadratic speedup over the classical counterpart.
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.
With the significant advancement in quantum computation during the past couple of decades, the exploration of machine-learning subroutines using quantum strategies has become increasingly popular. Gaussian process regression is a widely used technique in supervised classical machine learning. Here we introduce an algorithm for Gaussian process regression using continuous-variable quantum systems that can be realized with technology based on photonic quantum computers under certain assumptions regarding distribution of data and availability of efficient quantum access. Our algorithm shows that by using a continuous-variable quantum computer a dramatic speedup in computing Gaussian process regression can be achieved, i.e., the possibility of exponentially reducing the time to compute. Furthermore, our results also include a continuous-variable quantum-assisted singular value decomposition method of nonsparse low rank matrices and forms an important subroutine in our Gaussian process regression algorithm.
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.