[论文解读] A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

Emmanuel Jeandel, Simon Perdrix+1 more|arXiv (Cornell University)|May 31, 2017

Quantum Mechanics and Applications参考文献 37被引用 109

一句话总结

本文证明 ZX-Calculus 的 π/4 片段对于 Clifford+T 量子力学是完备的，通过将 ZX 与 ZW1/2-calculus 相关联并提供两个新的公理。同时，它将 ZXπ/4 的表达能力表征为 dyadic-extended 字段上的矩阵。

ABSTRACT

We introduce the first complete and approximatively 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 four new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that the Clifford+T fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.

研究动机与目标

• Motivation: to obtain a complete, approximately universal diagrammatic language for Clifford+T quantum mechanics.
• Goal: extend ZX-Calculus with two axioms to achieve completeness for the π/4 fragment.
• Approach: leverage the complete ZW1/2-calculus and back-and-forth translations between ZX and ZW1/2.
• Outcome: derive completeness of ZXπ/4 and characterize the exact expressive power of ZXπ/4 diagrams.

提出的方法

• Introduce two new ZX-Calculus axioms (C) and (BW) for the π/4 fragment.
• Define and extend the ZW-Calculus to ZW1/2 to express dyadic rational matrices.
• Provide two interpretations between ZXπ/4 and ZW1/2 and prove their correctness.
• Show that ZXπ/4 completeness follows from ZW1/2 completeness via back-and-forth translations.
• Prove that ZXπ/4 represents exactly matrices over the ring D[eiπ/4].
• Characterize the expressive power of ZXπ/4 in terms of dyadic rationals and roots of unity.

实验结果

研究问题

• RQ1 Can the π/4 fragment of the ZX-Calculus be made complete for Clifford+T quantum mechanics?
• RQ2 How can completeness be established by connecting ZX to a calculus with simpler numerical foundations (ZW1/2)?
• RQ3 What exact matrices are representable by ZXπ/4 diagrams?
• RQ4 What is the role and necessity of the two new axioms (C) and (BW) in achieving completeness?
• RQ5 How does the translation between ZXπ/4 and ZW1/2 preserve semantics and provability?

主要发现

• The π/4 fragment of ZX-Calculus is complete for Clifford+T quantum mechanics.
• A ZW1/2 calculus for dyadic rationals is complete and used as a bridge to prove ZXπ/4 completeness.
• Back-and-forth interpretations between ZXπ/4 and ZW1/2 preserve semantics and provability.
• ZXπ/4-diagrams represent exactly matrices over the field D[eiπ/4] (dyadic rationals extended by eiπ/4).
• There exists a completion procedure where each ZW1/2 axiom yields a ZX-equation, two new ZX axioms suffice for completeness.