Qufinite ZX-calculus: a unified framework of qudit ZX-calculi

Abstract

ZX-calculus is graphical language for quantum computing which usually focuses on qubits. In this paper, we generalise qubit ZX-calculus to qudit ZX-calculus in any finite dimension by introducing suitable generators, especially a carefully chosen triangle node. As a consequence we obtain a set of rewriting rules which can be seen as a direct generalisation of qubit rules, and a normal form for any qudit vectors. Based on the qudit ZX-calculi, we propose a graphical formalism called qufinite ZX-calculus as a unified framework for all qudit ZX-calculi, which is universal for finite quantum theory due to a normal form for matrix of any finite size. As a result, it would be interesting to give a fine-grained version of the diagrammatic reconstruction of finite quantum theory [Selby2021reconstructing] within the framework of qufinite ZX-calculus.

38 pages, changed the new qudit normal form which looks nicer now

References in corpus (2)

• Qutrit Dichromatic Calculus and Its Universality
• Completeness of the ZX-calculus

Cited by in corpus (7)

• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• The Qudit ZH-Calculus: Generalised Toffoli+Hadamard and Universality
• Compilation of Entangling Gates for High-Dimensional Quantum Systems
• A non-anyonic qudit ZW-calculus
• Scaling W state circuits in the qudit Clifford hierarchy
• Complete ZX-calculi for the stabiliser fragment in odd prime dimensions
• Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning