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

TL;DR

This paper extends the ZH calculus to qudits of prime-power dimensions, enabling graphical reasoning about finite field arithmetic, and demonstrates its universality and application to quantum algorithms like polynomial interpolation.

Contribution

It introduces a generalized ZH calculus for finite fields of prime-power dimension, expanding the graphical language's applicability to field arithmetic in quantum computing.

Findings

The generalized ZH calculus is universal over matrices with entries in $ ext{Z}[ ext{ω}]$.

Graphical description of a quantum polynomial interpolation algorithm.

Extension from prime fields to prime-power fields enhances the expressiveness of the ZH calculus.

Abstract

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 $\mathbb Z / q\mathbb Z$ 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.