Quantum Regularized Least Squares

Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning statistics. In most real-world scenarios, however, problems are often ill-posed or the underlying model suffers from overfitting, leading erroneous trivial solutions. This dealt with by adding extra constraints, known regularization. this paper, we use frameworks of block-encoding quantum singular value transformation (QSVT) design first algorithms for least squares general $\ell_2$-regularization. These include regularized versions ordinary squares, weighted generalized squares. Our substantially improve upon prior results on ridge (polynomial improvement in condition number an exponential accuracy), which particular case our result. To end, assume approximate block-encodings matrices input robust QSVT various algebra operations. particular, develop variable-time algorithm matrix inversion using QSVT, where eigenvalue discrimination subroutine instead gapped phase estimation. ensures that fewer ancilla qubits required procedure than results. Owing generality framework, applicable variety can also be seen improved standard (non-regularized) algorithms.