# A Simplified Stabilizer ZX-calculus

@inproceedings{Backens2016ASS, title={A Simplified Stabilizer ZX-calculus}, author={Miriam Backens and Simon Perdrix and Quanlong Wang}, booktitle={QPL}, year={2016} }

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics. We thus give a…

## 37 Citations

### 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.

### 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.

### Towards Minimality of Clifford + T ZX-Calculus

- Mathematics
- 2018

ZX-calculus is a high-level graphical language used for quantum computation. A complete set of axioms for ZX-calculus has been found very recently. This thesis works towards minimizing axioms of…

### Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions

- Computer ScienceMFCS
- 2022

A family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions are introduced, and it is proved that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations.

### 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.

### 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.

### Circuit Relations for Real Stabilizers: Towards TOF+H

- MathematicsArXiv
- 2019

This work completes the category CNOT generated by the controlled not gate and the computational ancillary bits, presented by circuit relations, to the real stabilizer fragment of quantum mechanics, and discusses how this could potentially lead to a complete axiomatization, in terms of circuit Relations, for the approximately universal fragment ofquantum mechanicsgenerated by the Toffoli gate, Hadamard gate and computational anCillary bits.

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

- PhysicsQPL
- 2018

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.

### 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.

### Y-Calculus: A language for real Matrices derived from the ZX-Calculus

- Mathematics, Computer ScienceQPL
- 2017

A ZX-like diagrammatic language devoted to manipulating real matrices - and rebits -, with its own set of axioms, is introduced, and it is proved that some restriction of the language is complete.

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