Interacting Quantum Observables: Categorical Algebra and Diagrammatics

Paper information (2011, New Journal of Physics 13, 043016; preprint: 2009) — Bob Coecke and Ross Duncan. DOI; arXiv:0906.4725

Work

The paper has two connected aims: an intuitive graphical calculus for multi-qubit systems and an axiomatization of complementary quantum observables in dagger symmetric monoidal categories. The ZX-calculus represents quantum processes with diagrammatic nodes and wires. Each observable is modeled by a commutative special dagger Frobenius algebra and gives rise to a phase group.

Contributions

• The ZX-calculus as a graphical language for quantum-computation derivations.
• A dagger symmetric monoidal account of complementarity and phase structure.
• A strong-complementarity and bialgebra account of the Z and X spin observables.
• A bridge between graphical calculus and categorical algebra for reasoning about quantum processes.

Limits

Graphical equations provide structured reasoning but do not automatically provide hardware compilation, a noise model, or efficient algorithms for every rewrite. An abstract category still needs a concrete model and scalar interpretation to describe a physical system; an expressible equation is not by itself evidence of hardware implementation or verification.

Connections

The paper develops the categorical foundations of A Categorical Semantics of Quantum Protocols and Dagger Compact Closed Categories and Completely Positive Maps, and supplies a language for graphical circuit reasoning. Compared with The Measurement Calculus, ZX-calculus emphasizes process diagrams and observable algebra; compared with Quantum Circuits, it works at a more abstract equational level.

Original paper

Bob Coecke and Ross Duncan, Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New Journal of Physics 13, 043016 (2011), DOI: 10.1088/1367-2630/13/4/043016. arXiv:0906.4725