Digital-Analog Co-Design of the Harrow-Hassidim-Lloyd Algorithm

Abstract

The Harrow-Hassidim-Lloyd quantum algorithm was proposed to solve linear systems of equations $A\stackrel{\ensuremath{\rightarrow}}{x}=\stackrel{\ensuremath{\rightarrow}}{b}$ 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 $A$ 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 $f(A)$ of matrix $A$, 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 $ϵ$ as $\mathcal{O}({ϵ}^{\ensuremath{-}1})$, 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 ``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.