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

@article{Jeandel2017ACA, title={A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics}, author={Emmanuel Jeandel and Simon Perdrix and Renaud Vilmart}, journal={Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science}, year={2017} }

We introduce the first complete and approximately universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics -- also called the π/4-fragment of the ZX-Calculus -- was one of the main open questions in categorical quantum mechanics. We prove the…

## 102 Citations

### Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

- PhysicsLICS
- 2018

It is shown 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 axiom atisation of the ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

### 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.

### Fe b 20 18 Diagrammatic Reasoning beyond Clifford + T Quantum Mechanics

- Physics
- 2018

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. An axiomatisation has recently been proven to be complete for an…

### Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

- Mathematics, Physics
- 2018

Recently, we gave a complete axiomatisation of the ZX-calculus [1] for the overall pure qubit quantum mechanics [4]. In this paper, we first simplify the rule of addition (AD) and show that some…

### 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.

### The rational fragment of the ZX-calculus

- MathematicsArXiv
- 2018

A new axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics, does not use any metarule, but relies instead on a more natural rule, called the cyclotomic supplementarity rule, that was introduced previously in the literature.

### A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

- PhysicsQPL
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A ZX-calculus augmented with triangle nodes is considered, and the form of the matrices it represents is precisely shown, and an axiomatisation is provided which makes the language complete for the Toffoli-Hadamard quantum mechanics.

### 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.

### Towards a Minimal Stabilizer ZX-calculus

- MathematicsLog. Methods Comput. Sci.
- 2020

It is shown that most of the remaining rules of the language are necessary, however leaving as an open question the necessity of two rules, including the bialgebra rule, which is an axiomatisation of complementarity, the cornerstone of the ZX-calculus.

### 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.

## 32 References

### Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

- PhysicsLICS
- 2018

It is shown 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 axiom atisation of the ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

### 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.

### Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

- Mathematics, Physics
- 2018

Recently, we gave a complete axiomatisation of the ZX-calculus [1] for the overall pure qubit quantum mechanics [4]. In this paper, we first simplify the rule of addition (AD) and show that some…

### A Simplified Stabilizer ZX-calculus

- PhysicsQPL
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It is proved that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the stabilizer ZX-calculus, can in fact be derived.

### Towards a Minimal Stabilizer ZX-calculus

- MathematicsLog. Methods Comput. Sci.
- 2020

It is shown that most of the remaining rules of the language are necessary, however leaving as an open question the necessity of two rules, including the bialgebra rule, which is an axiomatisation of complementarity, the cornerstone of the ZX-calculus.

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

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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.

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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.

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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".

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The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary…

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The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.