A Simplified Stabilizer ZX-calculus

TL;DR

This paper simplifies the stabilizer ZX-calculus by reducing axioms and deriving symmetries, making the language more fundamental and easier to work with in quantum reasoning.

Contribution

It provides a smaller, more essential set of axioms for the stabilizer ZX-calculus, removing unnecessary rules and demonstrating that certain symbols and rules are derivable.

Findings

Reduced the set of axioms for the stabilizer ZX-calculus

Derived symmetries previously considered as axioms

Showed scalar-tracking symbols are unnecessary

Abstract

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics. We thus give a significantly smaller set of axioms and prove that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the language, can in fact be derived. In particular, we show that the additional symbol and one of the rules which had been recently introduced to keep track of scalars (diagrams with no inputs or outputs) are not necessary.