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 $O({\kappa }^2\sqrt{N}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}(N)\mathrm{/}(ϵ\log \kappa ))$ for an $N\times N$dimensionalHermitianmatrix$\bm{F}$,where$\kappa$ denotes the condition number of $\mathit{F}$ and $ϵ$ 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 $\kappa$.