  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 $b \in \mathbb{C}^m$ be a vector. We
present two algorithms: a "sampling" algorithm that provides a sample from
$A^{-1}b$ and a "query" algorithm that outputs an estimate of an entry of
$A^{-1}b$, where $A^{-1}$ 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.
