Algebraic Structure of Quantum Controlled States and Operators

• URL: http://arxiv.org/abs/2603.13454v1
• Date: Fri, 13 Mar 2026 16:29:21 GMT
• Title: Algebraic Structure of Quantum Controlled States and Operators
• Abstract summary: We show that controlled diagrams in the ZXW-calculus admit rich structure.<n>The perspective of the higher-order map Ctrl recovers the standard notion of quantum controlled gates.
• Score: 0.6117371161379209
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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