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arXiv · 1801.10142

Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

Emmanuel Jeandel·Simon Perdrix·Renaud Vilmart
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Abstract

The ZX-Calculus is a graphical language for quantum mechanics. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clifford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structure, an essential feature of the ZX-calculus. In particular all the axioms of the ZX-calculus are involving linear diagrams. (ii) We also show that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiomatisation of the ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

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Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart. 2018-01-30. Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics. https://arxiv.org/abs/1801.10142

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