ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces

TL;DR

This paper proves the completeness of the finite-dimensional ZX-calculus, a graphical language for quantum computing, extending its applicability beyond qubits to all finite-dimensional quantum systems.

Contribution

It establishes the completeness of the ZX-calculus for all finite-dimensional Hilbert spaces using a new proof based on the ZW-calculus.

Findings

Completeness of ZX-calculus for finite-dimensional systems proven.

Direct translations between ZX and ZW calculi demonstrated.

Foundation laid for broader applications in quantum theory.

Abstract

The ZX-calculus is a graphical language for reasoning about quantum computing and quantum information theory. As a complete graphical language, it incorporates a set of axioms rich enough to derive any equation of the underlying formalism. While completeness of the ZX-calculus has been established for qubits and the Clifford fragment of prime-dimensional qudits, universal completeness beyond two-level systems has remained unproven until now. In this paper, we present a proof establishing the completeness of finite-dimensional ZX-calculus, incorporating only the mixed-dimensional Z-spider and the qudit X-spider as generators. Our approach builds on the completeness of another graphical language, the finite-dimensional ZW-calculus, with direct translations between these two calculi. By proving its completeness, we lay a solid foundation for the ZX-calculus as a versatile tool not only for quantum computation but also for various fields within finite-dimensional quantum theory.