Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

TL;DR

This paper introduces a complete diagrammatic calculus for qutrit stabilizer quantum mechanics, enabling derivation of all valid equations in this framework, which is more complex than the qubit case due to richer structure.

Contribution

It develops a qutrit ZX-calculus with rules different from the qubit version, proving its completeness for pure qutrit stabilizer quantum mechanics.

Findings

The qutrit ZX-calculus is complete for stabilizer quantum mechanics.

Diagrammatic derivations can reproduce all valid equations.

The structure is richer and more complex than the qubit case.

Abstract

In this paper, we show that a qutrit version of ZX-calculus, with rules significantly different from that of the qubit version, is complete for pure qutrit stabilizer quantum mechanics, where state preparations and measurements are based on the three dimensional computational basis, and unitary operations are required to be in the generalized Clifford group. This means that any equation of diagrams that holds true under the standard interpretation in Hilbert spaces can be derived diagrammatically. In contrast to the qubit case, the situation here is more complicated due to the richer structure of this qutrit ZX-calculus.