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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2406.02219 |
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| _version_ | 1866914253862076416 |
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| author | Gao, Dichuan |
| author_facet | Gao, Dichuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Qudit ZH Calculus for Arbitrary Finite Fields: Universality and Application Gao, Dichuan Quantum Physics 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. |
| title | The Qudit ZH Calculus for Arbitrary Finite Fields: Universality and Application |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2406.02219 |