The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification

Abstract

We present a smorgasbord of results on the stabiliser ZX-calculus for odd prime-dimensional qudits (i.e. qupits). We derive a simplified rule set that closely resembles the original rules of qubit ZX-calculus. Using these rules, we demonstrate analogues of the spider-removing local complementation and pivoting rules. This allows for efficient reduction of diagrams to the affine with phases normal form. We also demonstrate a reduction to a unique form, providing an alternative and simpler proof of completeness. Furthermore, we introduce a different reduction to the graph state with local Cliffords normal form, which leads to a novel layered decomposition for qupit Clifford unitaries. Additionally, we propose a new approach to handle scalars formally, closely reflecting their practical usage. Finally, we have implemented many of these findings in DiZX, a new open-source Python library for qudit ZX-diagrammatic reasoning.

In Proceedings QPL 2023, arXiv:2308.15489. 45 pages, lots of figures

References in corpus (9)

• High-dimensional quantum communication: benefits, progress, and future challenges
• High-Fidelity Qutrit Entangling Gates for Superconducting Circuits
• Native qudit entanglement in a trapped ion quantum processor
• Asymptotic Improvements to Quantum Circuits via Qutrits
• Diagonal gates in the Clifford hierarchy
• Decomposing the generalized Toffoli gate with qutrits
• Circuit QED: single-step realization of a multiqubit controlled phase gate with one microwave photonic qubit simultaneously controlling $n-1$ microwave photonic qubits
• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• Depicting qudit quantum mechanics and mutually unbiased qudit theories

Cited by in corpus (6)

• The Qudit ZH-Calculus: Generalised Toffoli+Hadamard and Universality
• Optimal compilation of parametrised quantum circuits
• A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits
• ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
• ZX Graphical Calculus for Continuous-Variable Quantum Processes
• Transversal AND in Quantum Codes