An improved quantum algorithm for low-rank rigid linear regressions with vector solution outputs

Abstract

Let A ∈ R n×d , b ∈ R n and λ > 0, for rigid linear regression arg minwe propose a quantum algorithm, in the framework of block-encoding, that returns a vector solution xopt such that Z(x opt ) ≤ (1 + ε)Z(x opt ), where x opt is an optimal solution. If a blockencoding of A is constructed in time O(T ), then the cost of the quantum algorithm is roughlyHere K = T α/λ and α is a normalization parameter such that A/α is encoded in a unitary through the block-encoding. This can be more efficient than naive quantum algorithms using quantum linear solvers and quantum tomography or amplitude estimation, which usually cost O(Kd/ε). The main technique we use is a quantum accelerated version of leverage score sampling, which may have other applications. The speedup of leverage score sampling can be quadratic or even exponential in certain cases. As a byproduct, we propose an improved randomized classical algorithm for rigid linear regressions. Finally, we show some lower bounds on performing leverage score sampling and solving linear regressions on a quantum computer.