2024/06/04 by Dichuan, Gao, Dichuan
Computer Science · Mathematics · #Chaos-based Image/Signal Encryption #Cryptography and Residue Arithmetic #FOS: Physical sciences #Quantum Physics (quant-ph) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2406.02219
openalex publication_date 2024/06/04 · openalex created_date 2024/06/08 · openalex updated_date 2026/07/28
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 \mathbb Z / q\mathbb Z rather than field arithmetic in \mathbb Fq. We show this generalized ZH calculus to be universal over matrices \mathbb Cqn → \mathbb Cqm with entries in the ring \mathbb 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.