Algebraic Structure of Quantum Controlled States and Operators

TL;DR

This paper explores the algebraic structure of quantum controlled states and operators within the ZXW-calculus, revealing rings and isomorphisms that enable advanced rewrite rules for quantum Hamiltonian factorization.

Contribution

It establishes that controlled matrices form a ring and controlled states are isomorphic to multilinear polynomials, providing a foundation for improved graphical calculus techniques.

Findings

Controlled matrices form a ring with rewrite rules.

Controlled states are isomorphic to multilinear polynomials.

Enables factorization of qubit Hamiltonians within graphical calculus.

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.