The Harrow-Hassidim-Lloyd quantum algorithm was proposed to solve linear systems of equations and it is the core of various applications. However, there is no explicit quantum circuit for the subroutine that maps the inverse of the problem matrix into an ancillary qubit. This makes implementation in current quantum devices challenging, forcing us to use hybrid approaches. Here, we propose a systematic method to implement this subroutine, which can be adapted to other functions of matrix , we present a co-designed quantum processor that reduces the depth of the algorithm, and we introduce its digital-analog implementation. The depth of our proposal scales with precision as , 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 “kitelike” 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.