Section: New Results

Completeness of the ZX-calculus

• Participants: Renaud Vilmart, Simon Perdrix, Emmanuel Jeandel

The ZX-Calculus is a powerful graphical language for quantum reasoning and quantum computing introduced by Bob Coecke and Ross Duncan [36]. The ZX-calculus has several applications in quantum information processing [37] (e.g. measurement-based quantum computing, quantum codes, foundations), and can be used through the interactive theorem prover Quantomatic. However, the main obstacle to wider use of the ZX-calculus was the absence of a completeness result for a universal fragment of quantum mechanics, in order to guarantee that any true property is provable using the ZX-calculus. We have introduced the first complete axiomatisation for a universal fragment of quantum mechanics. We also showed that a single additional rule makes the ZX-calculus complete for the whole pure qubit quantum mechanics. These results have been presented at LICS this year [16], [17] and will be presented at QIP'19, the main conference in quantum information processing.