Completeness of the ZX-calculus

• URL: http://arxiv.org/abs/2209.14894v2
• Date: Wed, 17 May 2023 17:48:27 GMT
• Title: Completeness of the ZX-calculus
• Authors: Quanlong Wang
• Abstract summary: We give the first complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. This paves the way for automated pictorial quantum computing, with the aid of some software like Quantomatic.
• Score: 0.3655021726150367
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The ZX-calculus is an intuitive but also mathematically strict graphical language for quantum computing, which is especially powerful for the framework of quantum circuits. Completeness of the ZX-calculus means any equality of matrices with size powers of $n$ can be derived purely diagrammatically. In this thesis, we give the first complete axiomatisation the ZX-calculus for the overall pure qubit quantum mechanics, via a translation from the completeness result of another graphical language for quantum computing -- the ZW-calculus. This paves the way for automated pictorial quantum computing, with the aid of some software like Quantomatic. Based on this universal completeness, we directly obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics, which is approximatively universal for quantum computing, by restricting the ring of complex numbers to its subring corresponding to the Clifford+T fragment resting on the completeness theorem of the ZW-calculus for arbitrary commutative ring. Furthermore, we prove the completeness of the ZX-calculus (with just 9 rules) for 2-qubit Clifford+T circuits by verifying the complete set of 17 circuit relations in diagrammatic rewriting. In addition to completeness results within the qubit related formalism, we extend the completeness of the ZX-calculus for qubit stabilizer quantum mechanics to the qutrit stabilizer system. Finally, we show with some examples the application of the ZX-calculus to the proof of generalised supplementarity, the representation of entanglement classification and Toffoli gate, as well as equivalence-checking for the UMA gate.

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