#  3D Quon Language 

 



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   ![Alex Wozniakowsi, Arthur Jaffe, and Zhengwei Liu](/sites/g/files/omnuum6611/files/styles/hwp_1_1__720x720_scale/public/mathpicture/files/alex_arthur_zhengwei.jpg?itok=Vm4DfN_5) 

 

**New Insights into Quantum Information**

We introduced a 3-dimensional language into quantum information, using charged strings in three-manifolds. For a 1-qudit transformation, we replace our 2-string model by a 4-string model. This allows us to use more powerful isotopy tools that extend the ideas in the 2-string model. In this language we can realize all Clifford gates as a single picture of a three-manifold with embedded charged strings. Consequently this model is universal for measurement-based quantum computation. Our model is linear, and we can realize all transformations as a linear sum of pictures. But we emphasize the use of single pictures and 3D isotopy.

**New Insights into Other Subjects**

We will update this soon**.**

Arthur Jaffe and Zhengwei Liu, A Mathematical Picture Language Program. [PNAS](http://www.pnas.org/content/115/1/81.full.pdf)

Zhengwei Liu, Quon Language: Surface Algebras and Fourier Duality. arXiv: [1707.03995](https://arxiv.org/abs/1707.03995)

Jacob Biamonte, Charged String Tensor Networks. The Biamonte paper is a commentary on, “Quon 3D Language for Quantum Information.” arXiv: [1703.01302](https://arxiv.org/abs/1703.01302); [PNAS](http://www.pnas.org/content/114/10/2447)

Zhengwei Liu, Alex Wozniakowski, and Arthur Jaffe. Quon 3D Language for Quantum Information. arXiv: [1612.02630](https://arxiv.org/pdf/1612.02630.pdf); [PNAS](http://www.pnas.org/content/114/10/2497.full.pdf)

   ![6-j Symbol Duality](/sites/g/files/omnuum6611/files/styles/hwp_1_1__720x720_scale/public/mathpicture/files/6j-dualitybig.jpg?itok=2tI4vJpj)