Algebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operations

Abstract

In this paper we give a complete axiomatisation of qubit ZX-calculus via elementary transformations which are basic operations in linear algebra. This formalism has two main advantages. First, all the operations of the phases are algebraic ones without trigonometry functions involved, thus paved the way for generalising complete axiomatisation of qubit ZX-calculus to qudit ZX-calculus and ZX-calculus over commutative semirings. Second, we characterise elementary transformations in terms of ZX diagrams, so a lot of linear algebra stuff can be done purely diagrammatically.

99 pages, many diagrams, added another normal form. arXiv admin note: substantial text overlap with arXiv:1912.01003

References in corpus (2)

• A universal completion of the ZX-calculus
• Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

Cited by in corpus (7)

• ZX-calculus for the working quantum computer scientist
• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• How to Sum and Exponentiate Hamiltonians in ZXW Calculus
• Completeness of the ZH-calculus
• A non-anyonic qudit ZW-calculus
• Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning
• Addition and Differentiation of ZX-diagrams