On Encoding Matrices using Quantum Circuits

Quantum Physics arXiv:2510.20030 (quant-ph) [Submitted on 22 Oct 2025] Title:On Encoding Matrices using Quantum Circuits Authors:Liron Mor Yosef, Haim Avron View a PDF of the paper titled On Encoding Matrices using Quantum Circuits, by Liron Mor Yosef and 1 other authors View PDF Abstract:Over a decade ago, it was demonstrated that quantum computing has the potential to revolutionize numerical linear algebra by enabling algorithms with complexity superior to what is classically achievable, e.g., the seminal HHL algorithm for solving linear systems. Efficient execution of such algorithms critically depends on representing inputs (matrices and vectors) as quantum circuits that encode or implement these inputs. For that task, two common circuit representations emerged in the literature: block encodings and state preparation circuits. In this paper, we systematically study encodings matrices in the form of block encodings and state preparation circuits. We examine methods for constructing these representations from matrices given in classical form, as well as quantum two-way conversions between circuit representations. Two key results we establish (among others) are: (a) a general method for efficiently constructing a block encoding of an arbitrary matrix given in classical form (entries stored in classical random access memory); and (b) low-overhead, bidirectional conversion algorithms between block encodings and state preparation circuits, showing that these models are essentially equivalent. From a technical perspective, two central components of our constructions are: (i) a special constant-depth multiplexer that simultaneously multiplexes all higher-order Pauli matrices of a given size, and (ii) an algorithm for performing a quantum conversion between a matrix's expansion in the standard basis and its expansion in the basis of higher-order Pauli matrices. Subjects: Quantum Physics (quant-ph); Machine Learning (cs.LG); Numerical Analysis (math.NA) Cite as: arXiv:2510.20030 [quant-ph] (or arXiv:2510.20030v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2510.20030 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Liron Mor Yosef [view email] [v1] Wed, 22 Oct 2025 21:20:08 UTC (260 KB) Full-text links: Access Paper: View a PDF of the paper titled On Encoding Matrices using Quantum Circuits, by Liron Mor Yosef and 1 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2025-10 Change to browse by: cs cs.LG cs.NA math math.NA References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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 Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)