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Physical Review A

Quantum algorithm for Gaussian process regression

Menghan Chen, Chao‐Hua Yu, Jianliang Gao, Kai Yu, Song Lin, Gongde Guo, Jing Li

Published Jul 5, 2022• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.106.012406

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In this paper, a quantum algorithm based on gaussian process regression model is proposed. The proposed quantum algorithm consists of three sub-algorithms. One is the first quantum subalgorithm to efficiently generate mean predictor. The improved HHL algorithm is proposed to obtain the sign of outcomes. Therefore, the terrible situation that results is ambiguous in terms of original HHL algorithm is avoided, which makes whole algorithm more clear and exact. The other is to product covariance predictor with same method. Thirdly, the squared exponential covariance matrices are prepared that annihilation operator and generation operator are simulated by the unitary linear decomposition Hamiltonian simulation and kernel function vectors is generated with blocking coding techniques on covariance matrices. In addition, it is shown that the proposed quantum gaussian process regression algorithm can achieve quadratic faster over the classical counterpart.
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.
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.
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.
Gaussian processes are probabilistic models that are commonly used as functional priors in machine learning. Due to their probabilistic nature, they can be used to capture prior information on the statistics of noise, smoothness of the functions, and training data uncertainty. However, their computational complexity quickly becomes intractable as the size of the data set grows. We propose a Hilbert-space approximation-based quantum algorithm for Gaussian process regression to overcome this limitation. Our method consists of a combination of classical basis function expansion with quantum computing techniques of quantum principal component analysis, conditional rotations, and Hadamard and tests. The quantum principal component analysis is used to estimate the eigenvalues, while the conditional rotations and the Hadamard and tests are employed to evaluate the posterior mean and variance of the Gaussian process. Our method provides polynomial computational complexity reduction over the classical method. Published by the American Physical Society 2024