The Qudit ZH Calculus for Arbitrary Finite Fields: Universality and Application
Dichuan Gao
TL;DR
This work extends the ZH graphical calculus to qudits of prime-power dimension $q=p^t$, implementing field multiplication in the finite field $\mathbb F_q$ and proving universality for linear maps whose matrix entries lie in $\mathbb Z[\omega]$ (with $\omega$ a primitive $p$-th root of unity). It introduces a phased finite-field ZH with Z-, H-, and $\xi$-state generators, using the field trace to realize the Fourier-like transform and to encode field arithmetic, while acknowledging a controlled loss of flexsymmetry in non-prime dimensions. Universality is shown by reducing arbitrary $R$-valued matrices to Schur products of pseudo-binary matrices and to polynomial-encoded diagrams $D_f$, enabling constructive diagrammatic representations of linear maps over $\mathbb F_q$. The polynomial-interpolation example demonstrates the calculus’s practicality for reasoning about quantum algorithms that rely on field arithmetic, highlighting a pathway to diagrammatic reasoning beyond prime dimensions while outlining future work on completeness and phase-free universality.
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 $\mathbb Z[ω]$ where $ω$ 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.