Optimization of the HHL Algorithm

Quantum Physics [Submitted on 16 Mar 2026] Optimization of the HHL Algorithm Dhruv Sood, Nilmani Mathur, Vikram Tripathi The Harrow-Hassidim-Lloyd (HHL) algorithm is a quantum algorithm for solving systems of linear equations that, in principle, offers an exponential improvement in scaling with the system size compared to classical approaches. In this work, we investigate the practical implementation and optimisation of the HHL algorithm with a focus on improving its performance on near-term quantum simulators. After outlining the algorithm, we examine two optimisation strategies aimed at improving fidelity and scalability: Suzuki-Trotter decomposition of the Hamiltonian evolution operator and a block-encoding approach that embeds the problem matrix into a larger unitary operator. The performance of these methods is evaluated through simulations on matrices with varying sparsity, including diagonal, tridiagonal, moderately dense, and fully dense cases. Our results show that while HHL achieves near-ideal fidelity for highly structured matrices, performance degrades as sparsity decreases due to the increasing cost of Hamiltonian simulation and reduced post-selection probability due to higher condition number. Block encoding is found to provide improved fidelity for moderately dense matrices, whereas Trotterisation offers a qubit-efficient approach for sparse systems. These results highlight the importance of matrix structure in determining the practical efficiency of HHL and inform future implementations that combine algorithmic optimisation with hardware-aware design. Comments: Proceedings of the 42nd International Symposium on Lattice Field Theory (Lattice 2025) Subjects: Quantum Physics (quant-ph); High Energy Physics - Lattice (hep-lat) Report number: TIFR/TH/26-10 Cite as: arXiv:2603.15756 [quant-ph]   (or arXiv:2603.15756v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.15756 Focus to learn more Submission history From: Dhruv Sood [view email] [v1] Mon, 16 Mar 2026 18:00:14 UTC (464 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 Change to browse by: hep-lat References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)