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 = pᵗ, 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 Z / q Z rather than field arithmetic in Fq. We show this generalized ZH calculus to be universal over matrices Cqⁿ → Cqᵐ with entries in the ring Z[ω] where ω is a pth 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.