  Solving linear systems is at the foundation of many algorithms. Recently,
quantum linear system algorithms (QLSAs) have attracted great attention since
they converge to a solution exponentially faster than classical algorithms in
terms of the problem dimension. However, low-complexity circuit implementations
of the oracles assumed in these QLSAs constitute the major bottleneck for
practical quantum speed-up in solving linear systems. In this work, we focus on
the application of QLSAs for linear systems that are expressed as a low rank
tensor sums, which arise in solving discretized PDEs. Previous works uses
modified Krylov subspace methods to solve such linear systems with a
per-iteration complexity being polylogarithmic of the dimension but with no
guarantees on the total convergence cost. We propose a quantum algorithm based
on the recent advances on adiabatic-inspired QLSA and perform a detailed
analysis of the circuit depth of its implementation. We rigorously show that
the total complexity of our implementation is polylogarithmic in the dimension,
which is comparable to the per-iteration complexity of the classical heuristic
methods.
