Do quantum linear solvers offer advantage for networks-based system of linear equations?

TL;DR

<3-5 sentence high-level summary>This paper investigates whether quantum linear solvers can outperform classical solvers for networks-based linear system problems (NLSPs) derived from graphs. It analyzes how the condition number, sparsity, and system-size growth of graph-derived matrices govern potential quantum advantage, focusing on HHL and several variants, plus a dream solver as a theoretical benchmark. Across 50 graph families (Laplacian and incidence-based), the authors classify families as best, better, good, or bad, finding exponential advantage to be rare and polynomial advantage more common, with improved QLS algorithms enhancing prospects for many graphs. They further generalize to a generalized hypercube superfamily, discuss qualitative predictors of advantage, and present limited quantum hardware demonstrations illustrating current practical constraints.

Abstract

In this exploratory numerical study, we assess the suitability of Quantum Linear Solvers (QLSs) toward providing a quantum advantage for Networks-based Linear System Problems (NLSPs). NLSPs naturally arise from graphs, and are of importance as they are connected to real-world applications. The achievable advantage with a QLS for an NLSP depends on the interplay between the scaling of condition number and sparsity of matrices associated with the graph family considered, as well as system size growth. We analyze 50 graph families and identify that within the scope of our study, only 4% of them exhibit prospects for an exponential advantage with the Harrow-Hassidim-Lloyd (HHL) algorithm relative to an efficient classical solver (best graphs), while about 20% of them show a polynomial advantage (better graphs). Furthermore, we report that some graph families graduate from offering no advantage with HHL to promising a polynomial advantage with improved algorithms such as the Childs-Kothari-Somma algorithm, while some other graph families exhibit futile exponential advantage. We introduce a unified graph superfamily and show the existence of infinite best and better graphs in it. We also conjecture the conditions under which one may visually examine a graph family and guess the prospects for an advantage. Finally, we very briefly touch upon some practical issues that may arise even if the aforementioned graph theoretic requirements are satisfied, including quantum hardware challenges.

Figures (38)

• Figure 1: An overview of the current study. (a) Illustration of the linear systems problem (where for simplicity, we assume real-valued entries for $A$, $\vec{b}$, and $\vec{x}$) and the runtime complexity scaling of quantum linear solvers and an efficient classical linear solver, for which we happen to borrow the runtime expression of the otherwise limited conjugate gradient method. Here, ${f}(s(\mathcal{N}))$ denotes a function of sparsity, $s(\mathcal{N})$, which in turn depends on the system size, $\mathcal{N}$. (b) Depiction of the connection between real-world applications, such as effective resistance determination and traffic flow congestion detection with graph Laplacian and graph incidence matrices respectively, and linear equations. (c) Schematic of our numerical survey on 50 graph families, where for each of them, we study $\kappa(\mathcal{N})$ and $s(\mathcal{N})$ behaviour with system size, $\mathcal{N}$ ($N$ for Laplacian matrix and $N+M$, for the incidence matrix) as well as system size scaling with $n$, where $n=1,2,\cdots$, to infer within the scope of our calculations if there is a potential for exponential advantage (best graph family), polynomial advantage (better graph family), sub-linear advantage (good graph family), or no advantage (bad graph family), all with the HHL algorithm and compared relative to the efficient classical linear solver.
• Figure 2: Data for the hypercube graph family (best graph family) from Laplacian matrix based systems: sub-figures showing (a) $\kappa$ and $s$ versus $N$, (b) $R$ versus $N$, as well as (c) $\tilde{R}$ versus $n$, and a reference linear curve, $n$.
• Figure 3: Better graph family from Laplacian matrix based system: sub-figures showing (a) $\kappa$ and $s$ versus $N$, (b) $R$ versus $N$, as well as (c) $\tilde{R}$ versus $n$ and a reference linear curve, $n$, for the modified Margulis-Gabber-Galil graph.
• Figure 4: Best graph family from incidence-matrix based system: sub-figures showing (a) $\kappa$ and $s$ versus $N'$, (b) $R$ versus $N'$, as well as (c) $\tilde{R}$ versus $n$ and a reference linear curve, $n$, for the directed hypercube graph.
• Figure 5: Better graph families of incidence-matrix based systems: The first, second and third column of each panel represents $\kappa$, $s$ vs. system size $N'=N+M$, $R(N')$ vs $N'$ and $\tilde{R}(n)$ vs. $n$ accompanied by a reference linear curve, $n$, for the first three better graph families listed in Table \ref{['tab:incidence_complexity']}.

Theorems & Definitions (41)

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