Algebraic Structure of Quantum Controlled States and Operators

Quantum Physics [Submitted on 13 Mar 2026] Algebraic Structure of Quantum Controlled States and Operators Edwin Agnew, Lia Yeh, Richie Yeung Quantum control is an important logical primitive of quantum computing programs, and an important concept for equational reasoning in quantum graphical calculi. We show that controlled diagrams in the ZXW-calculus admit rich algebraic structure. The perspective of the higher-order map Ctrl recovers the standard notion of quantum controlled gates, while respecting sequential and parallel composition and multiple-control. In this work, we prove that controlled square matrices form a ring and therefore satisfy powerful rewrite rules. We also show that controlled states form a ring isomorphic to multilinear polynomials. Putting these together, we have completeness for polynomials over same-size square matrices. These properties supply new rewrite rules that make factorisation of arbitrary qubit Hamiltonians achievable inside a single graphical calculus. Subjects: Quantum Physics (quant-ph); Logic in Computer Science (cs.LO) Cite as: arXiv:2603.13454 [quant-ph]   (or arXiv:2603.13454v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.13454 Focus to learn more Submission history From: Richie Yeung [view email] [v1] Fri, 13 Mar 2026 16:29:21 UTC (100 KB) Access Paper: view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 Change to browse by: cs cs.LO 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?)