Quantum-inspired sublinear classical algorithms for solving low-rank linear systems

Abstract: We present classical sublinear-time algorithms for solving low-rank linear systems of equations. Our algorithms are inspired by the HHL quantum algorithm for solving linear systems and the recent breakthrough by Tang of dequantizing the quantum algorithm for recommendation systems. Let $A \in \mathbb{C}^{m \times n}$ be a rank-$k$ matrix, and $b \in \mathbb{C}<sup>m$ be a vector. We present two algorithms: a "sampling" algorithm that provides a sample from $A<sup>{-1}b$ and a "query" algorithm that outputs an estimate of an entry of $A<sup>{-1}b$, where $A<sup>{-1}$ denotes the Moore-Penrose pseudo-inverse. Both of our algorithms have query and time complexity $$O(\mathrm{poly}(k, \kappa, |A|_F, 1/\epsilon)\,\mathrm{polylog}(m, n))$$, where $\kappa$ is the condition number of $A$ and $\epsilon$ is the precision parameter. Note that the algorithms we consider are sublinear time, so they cannot write and read the whole matrix or vectors. In this paper, we assume that $A$ and $b$ come with well-known low-overhead data structures such that entries of $A$ and $b$ can be sampled according to some natural probability distributions. Alternatively, when $A$ is positive semidefinite, our algorithms can be adapted so that the sampling assumption on $b$ is not required.