Rethinking Quantum Circuits

Quantum Physics arXiv:2608.19370 (quant-ph) [Submitted on 19 Aug 2026] Title:Rethinking Quantum Circuits Authors:Steven Rayan View a PDF of the paper titled Rethinking Quantum Circuits, by Steven Rayan View PDF HTML (experimental) Abstract:These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill--Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed. Comments: Subjects: Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Information Theory (cs.IT); Mathematical Physics (math-ph); Category Theory (math.CT) Cite as: arXiv:2608.19370 [quant-ph] (or arXiv:2608.19370v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2608.19370 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Steven Rayan [view email] [v1] Wed, 19 Aug 2026 18:39:27 UTC (1,588 KB) Full-text links: Access Paper: View a PDF of the paper titled Rethinking Quantum Circuits, by Steven RayanView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-08 Change to browse by: cond-mat cond-mat.mes-hall cs cs.IT math math-ph math.CT math.IT math.MP 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?) 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?)