Simplification Strategies for the Qutrit ZX-Calculus

Abstract: The ZX-calculus is a graphical language for suitably represented tensor networks, called ZX-diagrams. Calculations are performed by transforming ZX-diagrams with rewrite rules. The ZX-calculus has found applications in reasoning about quantum circuits, condensed matter systems, quantum algorithms, quantum error correcting codes, and counting problems. A key notion is the stabiliser fragment of the ZX-calculus, a subfamily of ZX-diagrams for which rewriting can be done efficiently in terms of derived simplifying rewrites. Recently, higher dimensional qudits - in particular, qutrits - have gained prominence within quantum computing research. The main contribution of this work is the derivation of efficient rewrite strategies for the stabiliser fragment of the qutrit ZX-calculus. Notably, this constitutes a first non-trivial step towards the simplification of qutrit quantum circuits. We then give further unexpected areas in which these rewrite strategies provide complexity-theoretic insight; namely, we reinterpret known results about evaluating the Jones polynomial, an important link invariant in knot theory, and counting graph colourings.