# The ZX calculus is incomplete for quantum mechanics

@inproceedings{Witt2014TheZC, title={The ZX calculus is incomplete for quantum mechanics}, author={C. S. D. Witt and Vladimir Zamdzhiev}, booktitle={QPL}, year={2014} }

We prove that the ZX-calculus is incomplete for quantum mechanics. We suggest the addition of a new 'color-swap' rule, of which currently no analytical formulation is known and which we suspect may be necessary, but not sufficient to make the ZX-calculus complete.

## 36 Citations

### Supplementarity is Necessary for Quantum Diagram Reasoning

- PhysicsMFCS
- 2016

It is proved that its π 4-fragment is not complete, in other words the ZX-calculus is notcomplete for the so called "Clifford+T quantum mechanics".

### A universal completion of the ZX-calculus

- MathematicsArXiv
- 2017

A universal completion of the ZX-calculus for the whole of pure qubit quantum mechanics is given, based on the completeness of another graphical language: the ZW-Calculus, with direct translations between these two graphical systems.

### The ZX Calculus is incomplete for Clifford+T quantum mechanics

- Physics
- 2015

The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. We prove that the ZX-calculus is not complete for Clifford+T quantum mechanics. The completeness…

### A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

- MathematicsLICS
- 2018

The ZX-Calculus is made complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language, and it is proved that the π/4-fragment of the ZX -Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.

### Generalised Supplementarity and new rule for Empty Diagrams to Make the ZX-Calculus More Expressive

- MathematicsArXiv
- 2017

To prove the incompleteness of the π 4-fragment, an equality over scalars that is not derivable from the set of axioms is shown, and that will lead to a substitute for the inverse rule, and the obsolescence of the zero rule.

### The ZX-Calculus is a powerful graphical language for quantum reasoning and quantum computing introduced

- Computer Science
- 2019

This work improves on the known-to-be-complete presentation for the so-called Clifford fragment of the ZX-Calculus and provides a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.

### A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics

- Mathematics2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2019

This paper introduces the singular-value decomposition of a ZX-diagram, and uses it to show that all the rules of the former axiomatisation are provable with the new one.

### Completeness of the ZX-Calculus

- MathematicsLog. Methods Comput. Sci.
- 2020

This work improves on the known-to-be-complete presentation for the so-called Clifford fragment of the ZX-Calculus, and provides a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.

### ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics

- MathematicsMFCS
- 2017

It is shown that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning, implying the incompleteness of the language for the so-called Clifford+T quantum mechanics.

### Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

- Mathematics
- 2018

In this paper, we show that a qutrit version of ZX-calculus, with rules significantly different from that of the qubit version, is complete for pure qutrit stabilizer quantum mechanics, where state…

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