The Qudit ZH Calculus for Arbitrary Finite Fields: Universality and Application

We propose a generalization of the graphical ZH calculus to qudits of prime-power dimensions $q=p^t$, implementing field arithmetic in arbitrary finite fields. This is an extension of a previous result by Roy which implemented arithmetic of prime-sized fields; and an alternative to a result by de Beaudrap which extended the ZH to implement cyclic ring arithmetic in $\mathbb{Z}\mathrm{/}q\mathbb{Z}$ rather than field arithmetic in ${\mathbb{F}}_q$. We show this generalized ZH calculus to be universal over matrices ${\mathbb{C}}^{q^n}\to {\mathbb{C}}^{q^m}$ with entries in the ring $\mathbb{Z}[\omega ]$ where $\omega$ is a $p$th root of unity. As an illustration of the necessity of such an extension of ZH for field rather than cyclic ring arithmetic, we offer a graphical description and proof for a quantum algorithm for polynomial interpolation. This algorithm relies on the invertibility of multiplication, and therefore can only be described in a graphical language that implements field, rather than ring, multiplication.