  The Harrow-Hassidim-Lloyd quantum algorithm was proposed to solve linear
systems of equations $A\vec{x} = \vec{b}$ and it is the core of various
applications. However, there is not an explicit quantum circuit for the
subroutine which maps the inverse of the problem matrix $A$ into an ancillary
qubit. This makes challenging the implementation in current quantum devices,
forcing us to use hybrid approaches. Here, we propose a systematic manner to
implement this subroutine, which can be adapted to other functions $f(A)$ of
the matrix $A$, we present a co-designed quantum processor which reduces the
depth of the algorithm, and we introduce its digital-analog implementation. The
depth of our proposal scales with the precision $\epsilon$ as
$\mathcal{O}(\epsilon^{-1})$, which is bounded by the number of samples allowed
for a certain experiment. The co-design of the Harrow-Hassidim-Lloyd algorithm
leads to a "kite-like" architecture, which allows us to reduce the number of
required SWAP gates. Finally, merging a co-design quantum processor
architecture with a digital-analog implementation contributes to the reduction
of noise sources during the experimental realization of the algorithm.
