Completeness of algebraic ZX-calculus over arbitrary commutative rings and semirings

Abstract

ZX-calculus is a strict mathematical formalism for graphical quantum computing which is based on the field of complex numbers. In this paper, we extend its power by generalising ZX-calculus to such an extent that it is universal both in an arbitrary commutative ring and in an arbitrary commutative semiring. Furthermore, we follow the framework of arXiv:2007.13739 to prove respectively that the proposed ZX-calculus over an arbitrary commutative ring (semiring) is complete for matrices over the same ring (semiring), via a normal form inspired from matrix elementary operations such as row addition and row multiplication. This work could lead to various applications including doing elementary number theory in string diagrams.

arXiv comment73 pages

References in corpus (2)

• The algebra of entanglement and the geometry of composition
• Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

Cited by in corpus (5)

• ZX-calculus for the working quantum computer scientist
• Qufinite ZX-calculus: a unified framework of qudit ZX-calculi
• Algebraic complete axiomatisation of ZX-calculus with a normal form via elementary matrix operations
• Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning
• Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)