Quantum Data Fitting Algorithm for Non-sparse Matrices

We propose a quantum data fitting algorithm for non-sparse matrices, which is based on the Quantum Singular Value Estimation (QSVE) subroutine and a novel efficient method for recovering the signs of eigenvalues. Our algorithm generalizes the quantum data fitting algorithm of Wiebe, Braun, and Lloyd for sparse and well-conditioned matrices by adding a regularization term to avoid the over-fitting problem, which is a very important problem in machine learning. As a result, the algorithm achieves a sparsity-independent runtime of $O(\kappa^2\sqrt{N}\mathrm{polylog}(N)/(\epsilon\log\kappa))$ for an $N\times N$ dimensional Hermitian matrix $\bm{F}$, where $\kappa$ denotes the condition number of $\bm{F}$ and $\epsilon$ is the precision parameter. This amounts to a polynomial speedup on the dimension of matrices when compared with the classical data fitting algorithms, and a strictly less than quadratic dependence on $\kappa$.