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

• URL: http://arxiv.org/abs/2406.02219v1
• Date: Tue, 4 Jun 2024 11:21:10 GMT
• Title: The Qudit ZH Calculus for Arbitrary Finite Fields: Universality and Application
• Authors: Dichuan, Gao,
• Abstract summary: We propose a generalization of the graphical ZH calculus to qudits of prime-power dimensions $q = pt$. We show this calculus to be universal over matrices $mathbb Cqn to mathbb Cqm$ with entries in the ring $mathbb Z[omega]$ where $omega$ is a $p$th root of unity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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