  Ridge regression (RR) is an important machine learning technique which
introduces a regularization hyperparameter $\alpha$ to ordinary multiple linear
regression for analyzing data suffering from multicollinearity. In this paper,
we present a quantum algorithm for RR, where the technique of parallel
Hamiltonian simulation to simulate a number of Hermitian matrices in parallel
is proposed and used to develop a quantum version of $K$-fold cross-validation
approach, which can efficiently estimate the predictive performance of RR. Our
algorithm consists of two phases: (1) using quantum $K$-fold cross-validation
to efficiently determine a good $\alpha$ with which RR can achieve good
predictive performance, and then (2) generating a quantum state encoding the
optimal fitting parameters of RR with such $\alpha$, which can be further
utilized to predict new data. Since indefinite dense Hamiltonian simulation has
been adopted as a key subroutine, our algorithm can efficiently handle
non-sparse data matrices. It is shown that our algorithm can achieve
exponential speedup over the classical counterpart for (low-rank) data matrices
with low condition numbers. But when the condition numbers of data matrices is
large to be amenable to full or approximately full ranks of data matrices, only
polynomial speedup can be achieved.
