Quantum algorithm for Gaussian process regression

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.

Figure 1

The quantum circuit of the kernel vector. Here the $H$ represents the Hadamard transform, $U$ and $U^{†}$ denote preparation for the coherent states of step 1.2 and the inverse process of step 1.7, respectively, and $R$ is the controlled rotation of step 1.6.

Figure 2

The preparation circuit of the predictive mean value and variance value. Here $PE$ and $P{E}^{†}$ denote the phase estimation algorithm and inverse phase estimation algorithm of steps B2 and B3, respectively. The $R^{\prime }$ is controlled rotation of step B3. SWAP is the swap operation of step B4.