A shortcut to an optimal quantum linear system solver

[2406.12086] A shortcut to an optimal quantum linear system solver Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Quantum Physics arXiv:2406.12086 (quant-ph) [Submitted on 17 Jun 2024 ( v1 ), last revised 8 Apr 2026 (this version, v2)] Title: A shortcut to an optimal quantum linear system solver Authors: Alexander M. Dalzell View a PDF of the paper titled A shortcut to an optimal quantum linear system solver, by Alexander M. Dalzell View PDF HTML (experimental) Abstract: Given a linear system of equations $A\boldsymbol{x}=\boldsymbol{b}$, quantum linear system solvers (QLSSs) approximately prepare a quantum state $|\boldsymbol{x}\rangle$ for which the amplitudes are proportional to the solution vector $\boldsymbol{x}$. Asymptotically optimal QLSSs have query complexity $O(\kappa \log(1/\varepsilon))$, where $\kappa$ is the condition number of $A$, and $\varepsilon$ is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm $\lVert\boldsymbol{x}\rVert$ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is $(1+O(\varepsilon))\kappa \ln(2\sqrt{2}/\varepsilon)$. If the norm is unknown, our method allows it to be estimated up to a constant factor using $O(\log\log(\kappa))$ applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal $O(\kappa \log\log(\kappa)\log\log\log(\kappa)+\kappa\log(1/\varepsilon))$ total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that $O(\kappa)$ complexity can be achieved for norm estimation, yielding an optimal QLSS with $O(\kappa\log(1/\varepsilon))$ complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of $56\kappa+1.05\kappa \ln(1/\varepsilon)+o(\kappa)$ for the complexity of our optimal QLSS. Comments: 13 pages main, 42 pages including appendices. v2: fixed typos, added citations and clarifications, added Appendix A.6 and Appendix G Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2406.12086 [quant-ph] (or arXiv:2406.12086v2 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2406.12086 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexander Dalzell [ view email ] [v1] Mon, 17 Jun 2024 20:54:11 UTC (313 KB) [v2] Wed, 8 Apr 2026 23:55:31 UTC (303 KB) Full-text links: Access Paper: View a PDF of the paper titled A shortcut to an optimal quantum linear system solver, by Alexander M. Dalzell View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2024-06 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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