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arXiv:1803.10520 · Physical Review A

Quantum algorithms for training Gaussian processes

Zhikuan Zhao, Jack Fitzsimons, Michael A. Osborne, Stephen Roberts, Joseph F. Fitzsimons

Published Jul 8, 2019• 19 citations
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Abstract

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 ($\mathrm{LML}$) of a given model. The $\mathrm{LML}$ gives a concrete measure of the quality of prediction that a GP model is expected to achieve. The classical computation of the $\mathrm{LML}$ 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 $\mathbf{y}$ is a dense vector and $A$ is the covariance matrix. We hence show that quantum computing can be used to estimate the $\mathrm{LML}$ of a GP with exponentially improved efficiency under certain conditions.

DOI
10.1103/physreva.100.012304

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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 yTA\ensuremath−1y{\mathbf{y}}^{T}{A}^{\ensuremath{-}1}\mathbf{y}yTA\ensuremath−1y, 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.
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 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 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.
The logarithm-determinant is a widely-present operation in many areas of physics and computer science. Derivatives of the logarithm-determinant compute physically relevant quantities in statistical physics models, quantum field theories, as well as the inverses of matrices. A multi-variable version of the quantum gradient algorithm is developed here to evaluate the derivative of the logarithm-determinant. From this, the pseudo-inverse of a sparse-rank input operator may be determined efficiently. Measuring an expectation value of the quantum state--instead of all N2N^2N2 elements of the input operator--can be accomplished in O(k/ε2)O(k/\varepsilon^2)O(k/ε2) time in the idealized case for kkk relevant eigenvectors of the input matrix with precision ε\varepsilonε. A practical implementation of the required operator will likely need log⁡2N\log_2Nlog2​N overhead, giving an overall complexity of O((klog⁡2N)/ε2)O((k\log_2 N)/\varepsilon^2)O((klog2​N)/ε2). The method applies widely and converges super-linearly in kkk when the condition number is high. The best classical method we are aware of scales as NNN. Given the same resource assumptions as other algorithms, such that an equal superposition of eigenvectors is available efficiently, the algorithm is evaluated in the practical case as O(log⁡2N/ε2)O(\log_2 N/\varepsilon^2)O(log2​N/ε2). The output is given in O(1)O(1)O(1) queries of an oracle, which is given explicitly here and only relies on time-evolution operators that can be implemented with arbitrarily small error. The algorithm is envisioned for fully error-corrected quantum computers but may be implementable on near-term machines. We discuss how this algorithm can be used for kernel-based quantum machine-learning.
In this thesis, I present several results on quantum statistical inference in the following two directions. Firstly, I demonstrate that quantum algorithms can be applied to enhance the computing and training of Gaussian processes (GPs), a powerful model widely used in classical statistical inference and supervised machine learning. A crucial component of the quantum GP algorithm is solving linear systems with quantum computers, for which I present a novel algorithm that achieves a provable advantage over previously known methods. I will also explicitly address the task of encoding the classical data into a quantum state for machine learning applications. I then apply the quantum enhanced GPs to Bayesian deep learning and present an experimental demonstration on contemporary hardware and simulators. Secondly, I look into the notion of quantum causality and apply it to inferring spatial and temporal quantum correlations, and present an analytical toolkit for causal inference in quantum data. I will also make the connection between causality and quantum communications, and present a general bound for the quantum capacity of noisy communication channels.