Algebraic Structure of Quantum Controlled States and Operators
This work addresses the need for algebraic structure in quantum control for quantum computing programs and graphical calculi, offering incremental improvements in equational reasoning.
The paper tackles the problem of representing quantum controlled states and operators algebraically by proving that controlled square matrices form a ring and controlled states form a ring isomorphic to multilinear polynomials, enabling completeness for polynomials over same-size square matrices. This result provides new rewrite rules that make factorisation of arbitrary qubit Hamiltonians achievable within a single graphical calculus.
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.