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

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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.