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.

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

References in corpus (36)

• Quantum cryptography: Public key distribution and coin tossing
• Improved Simulation of Stabilizer Circuits
• Qudits and high-dimensional quantum computing
• High-dimensional quantum communication: benefits, progress, and future challenges
• A universal qudit quantum processor with trapped ions
• Interacting Quantum Observables: Categorical Algebra and Diagrammatics
• Quantum Information Scrambling in a Superconducting Qutrit Processor
• Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus
• Enhanced fault-tolerant quantum computing in $d$-level systems
• The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
• Reducing T-count with the ZX-calculus
• High-Fidelity Qutrit Entangling Gates for Superconducting Circuits
• The ZX-calculus is complete for stabilizer quantum mechanics
• Native qudit entanglement in a trapped ion quantum processor
• Factoring with Qutrits: Shor's Algorithm on Ternary and Metaplectic Quantum Architectures
• PyZX: Large Scale Automated Diagrammatic Reasoning
• Asymptotic Improvements to Quantum Circuits via Qutrits
• The ZX calculus is a language for surface code lattice surgery
• There and back again: A circuit extraction tale
• Diagonal gates in the Clifford hierarchy
• Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus
• ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity
• Quantum Natural Language Processing on Near-Term Quantum Computers
• Decomposing the generalized Toffoli gate with qutrits
• Verifying the Steane code with Quantomatic
• Making the stabilizer ZX-calculus complete for scalars
• Verifying the Smallest Interesting Colour Code with Quantomatic
• Circuit QED: single-step realization of a multiqubit controlled phase gate with one microwave photonic qubit simultaneously controlling $n-1$ microwave photonic qubits
• Interacting Hopf Algebras: the theory of linear systems
• A Simplified Stabilizer ZX-calculus
• Efficient quantum gate teleportation in higher dimensions
• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics
• Depicting qudit quantum mechanics and mutually unbiased qudit theories
• Supplementarity is Necessary for Quantum Diagram Reasoning
• Well-tempered ZX and ZH Calculi

Cited by in corpus (7)

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