Stacy Patterson

Solving Laplacian linear systems Lx = b is central to many problems in network dynamics, including heat diffusion, distributed consensus, opinion formation, and centrality estimation. We study three quantum approaches to this problem: Harrow–Hassidim–Lloyd (HHL), the Variational Quantum Linear Solver (VQLS), and Quantum Singular Value Transformation (QSVT). Using a distributed consensus model on graph Laplacians, we compare these methods on ideal statevector simulation and validate HHL on IBM's ibm_rensselaer backend. Our results show that VQLS achieves the best simulated accuracy on small, well-conditioned instances, with relative l2 error as low as 1.18%, while HHL and QSVT exhibit larger errors under the tested settings. On hardware, HHL reaches 18.96% error after backend-specific parameter tuning, indicating that near-term performance is strongly affected by device noise and circuit depth.