Do quantum linear solvers offer advantage for networks-based system of linear equations? (2509.00913v1)

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 are of importance as they are naturally connected to real-world applications. In an NLSP, one starts with a graph and arrives at a system of linear equations. The advantage that one may obtain with a QLS for an NLSP is determined by the interplay between three variables: the scaling of condition number and sparsity functions of matrices associated with the graphs considered, as well as the function describing the system size growth. We recommend graph families that can offer potential for an exponential advantage (best graph families) and those that offer sub-exponential but at least polynomial advantage (better graph families), with the HHL algorithm considered relative to the conjugate gradient (CG) method. Within the scope of our analyses, we observe that only 4 percent of the 50 considered graph families offer prospects for an exponential advantage, whereas about 20 percent of the considered graph families show a polynomial advantage. Furthermore, we observe and report some interesting cases where some graph families not only fare better with improved algorithms such as the Childs-Kothari-Somma algorithm but also graduate from offering no advantage to promising a polynomial advantage, graph families that exhibit futile exponential advantage, etc. Given the limited number of graph families that one can survey through numerical studies, we discuss an interesting case where we unify several graph families into one superfamily, and show the existence of infinite best and better graphs in it. Lastly, we very briefly touch upon some practical issues that one may face even if the aforementioned graph theoretic requirements are satisfied, including quantum hardware challenges.

• The paper demonstrates that only select graph families, like the hypercube, offer exponential quantum advantage through favorable scaling of condition number and sparsity.
• The paper compares various QLS algorithms, including HHL variants and improved methods, by quantifying runtime complexity ratios and crossover points against classical approaches.
• The paper highlights key challenges such as state preparation, edge weight sensitivity, and hardware limitations that impact the practical realization of quantum advantage.

Quantum Linear Solvers for Network-Based Systems: Numerical Assessment of Quantum Advantage

Introduction and Motivation

This work presents a comprehensive numerical paper of quantum linear solvers (QLSs) applied to network-based linear system problems (NLSPs), focusing on the potential for quantum advantage over classical algorithms such as the conjugate gradient (CG) method. NLSPs are ubiquitous in scientific and engineering domains, arising naturally from graph-theoretic models of real-world systems (e.g., electrical circuits, traffic networks). The paper systematically analyzes the scaling behavior of key matrix properties—condition number ($\kappa$), sparsity ($s$), and system size ($\mathcal{N}$)—across 50 graph families, evaluating the prospects for exponential, polynomial, or sub-linear quantum speedup.

Figure 1: Overview of the paper, including the mapping from graphs to linear systems, runtime complexity scaling for QLSs and CG, and the survey workflow for graph families.

Quantum Linear Solvers: Algorithms and Complexity

The prototypical QLS is the Harrow-Hassidim-Lloyd (HHL) algorithm, with runtime complexity $O(\log(\mathcal{N}) s^2 \kappa^3 / \epsilon)$, where $\epsilon$ is the target precision. Several improved QLSs are considered, including HHL variants (Amplitude Amplification, Variable Time Amplitude Amplification, Psi-HHL), the Childs-Kothari-Somma (CKS) algorithm, adiabatic approaches, and a hypothetical "dream QLS" with optimal scaling. All QLSs offer at least a logarithmic dependence on system size, in contrast to the linear scaling of CG.

The paper introduces the runtime complexity ratio $$R(\mathcal{N}) = t_{\mathrm{CG}} / t_{\mathrm{QLS}}$$ and its extrapolated form $\tilde{R}(n)$, which serve as quantitative metrics for quantum advantage. The scaling of $\kappa$ and $s$ with $\mathcal{N}$, as well as the growth rate of $\mathcal{N}$ itself, are critical determinants of practical speedup.

Graph-Theoretic Formulation of NLSPs

NLSPs are constructed from graph Laplacian or incidence matrices, leading to systems of the form $L\vec{x} = \vec{b}$ or $B\vec{x} = \vec{b}$, respectively. The paper considers both undirected and directed graphs, with and without source/sink vertices, and surveys a diverse set of graph families (random, regular, trees, expanders, hypercubes, etc.). The classification of graph families is based on the degree of quantum advantage achievable with HHL:

• Best graphs: Exponential advantage
• Better graphs: Polynomial advantage
• Good graphs: Sub-linear advantage
• Bad graphs: No advantage

Numerical Survey: Results and Analysis

Laplacian Matrix-Based Systems

Out of 30 graph families, only the hypercube graph exhibits exponential quantum advantage, with both $\kappa$ and $s$ scaling logarithmically and system size growing exponentially ($N \sim 2^n$).

Figure 2: Hypercube graph—(a) $\kappa$ and $s$ vs $N$, (b) $R$ vs $N$, (c) $\tilde{R}$ vs $n$.

Two families (modified Margulis-Gabber-Galil and Sudoku graphs) yield polynomial advantage, with polylogarithmic scaling of $\kappa$ and $s$ but only polynomial system size growth.

Figure 3: Modified Margulis-Gabber-Galil and Sudoku graphs—scaling of $\kappa$, $s$, and runtime ratios.

Six families are classified as "good" (sub-linear advantage), typically due to linear system size growth despite favorable $\kappa$ and $s$ scaling. The majority of surveyed graphs are "bad," with either $\kappa$ or $s$ scaling polynomially, precluding quantum advantage.

Incidence Matrix-Based Systems

Among 20 directed graph families, only the directed hypercube graph achieves exponential advantage ($\mathcal{N} \sim n2^n$, $\kappa, s \sim \mathrm{polylog}(\mathcal{N})$).

Figure 4: Directed hypercube graph—(a) $\kappa$ and $s$ vs $N'$, (b) $R$ vs $N'$, (c) $\tilde{R}$ vs $n$.

Nine families offer polynomial advantage, with sub-exponential but super-linear system size growth. Four families are "good," and six are "bad," mirroring the Laplacian case.

Algorithmic Trade-offs and Crossover Points

Improved QLSs such as CKS and AQC algorithms consistently achieve quantum advantage at smaller system sizes compared to HHL, due to more favorable scaling in $\kappa$ and $\epsilon$. The paper quantifies crossover points for $R(\mathcal{N}) > 1$ across graph families, highlighting the practical superiority of these algorithms.

Edge Weight and Parameter Sensitivity

The scaling of edge weights can dramatically affect $\kappa$ and thus the potential for quantum advantage. For example, hypercube graphs retain exponential advantage with constant or logarithmic edge weights but lose it with linear or polynomial edge weights.

Figure 5: Edge weight analysis for hypercube graphs—logarithmic, linear, and polynomial edge weight functions.

Generalized Hypercube Superfamily

The paper introduces the generalized hypercube graph superfamily, parameterized by $(a, m)$, and demonstrates the existence of infinite best and better graph families within this framework. Rows, columns, and diagonals of the tableau correspond to families with polynomial or exponential system size growth and favorable $\kappa$, $s$ scaling.

Figure 6: Tableau of generalized hypercube graph families, with $\kappa$ and $s$ annotated for each cell.

Practical and Theoretical Implications

Hardware and Implementation Challenges

Even with favorable graph-theoretic properties, practical realization of quantum advantage is impeded by state preparation costs, feature extraction limitations, and hardware constraints. NISQ-era devices are limited to very small system sizes, as demonstrated by proof-of-concept HHL runs on IonQ hardware for $(4 \times 4)$ Laplacian matrices.

Figure 7: Schematic of the HHL algorithm with feature extraction via the HOM module.

Figure 8: Minimum eigenvalue threshold analysis for numerical stability in condition number estimation.

Algorithmic Considerations

Efficient state preparation and feature extraction are application-dependent and remain open problems. The extraction of the full solution vector $\vec{x}$ negates quantum advantage; only specific features (e.g., overlaps, effective resistance) can be efficiently computed.

Scaling and Resource Requirements

The paper emphasizes that exponential advantage requires not only polylogarithmic scaling of $\kappa$ and $s$ but also exponential system size growth. Polynomial or sub-linear advantage may be insufficient in the fault-tolerant era due to error correction overheads, as highlighted in prior work.

Future Directions

• Broader graph family surveys: Extending numerical analysis to additional graph superfamilies and parameter regimes.
• Semi-empirical spectral formulas: Developing analytical or semi-empirical models for $\kappa$ and $s$ scaling in complex networks.
• Algorithmic innovation: Designing QLSs with improved state preparation and feature extraction protocols.
• Hardware co-design: Integrating graph-theoretic insights with quantum hardware capabilities for practical NLSP applications.

Conclusion

This paper provides a rigorous numerical assessment of quantum linear solvers for network-based systems, identifying the structural and scaling prerequisites for quantum advantage. Exponential speedup is rare, with only 4% of surveyed graph families qualifying, while polynomial advantage is more common but still limited. The results underscore the necessity of favorable scaling in all relevant parameters and highlight the interplay between graph structure, algorithmic complexity, and hardware constraints. The generalized hypercube superfamily offers a promising avenue for future exploration of advantageous graph constructions. Realizing practical quantum advantage in NLSPs will require continued progress in both algorithmic and hardware domains, informed by detailed graph-theoretic analysis.