How to check universality of quantum gates?

• URL: http://arxiv.org/abs/2111.03862v6
• Date: Fri, 17 Jun 2022 13:48:06 GMT
• Title: How to check universality of quantum gates?
• Authors: Adam Sawicki, Lorenzo Mattioli and Zolt\'an Zimbor\'as
• Abstract summary: Our first criterion states that $mathcalSsubset G_d:=U(d)$ is universal if and only if $mathcalS$ forms a $delta$-approximate $t(d)$-design. Our second universality criterion says that $mathcalSsubset G_d$ is universal if and only if the centralizer of $mathcalSt(d),t(d)=Uotimes t(d)otimes t(d)|U
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We provide two simple universality criteria. Our first criterion states that $\mathcal{S}\subset G_d:=U(d)$ is universal if and only if $\mathcal{S}$ forms a $\delta$-approximate $t(d)$-design, where $t(2)=6$ and $t(d)=4$ for $d\geq3$. Our second universality criterion says that $\mathcal{S}\subset G_d$ is universal if and only if the centralizer of $\mathcal{S}^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in \mathcal{S}\}$ is equal to the centralizer of $G_d^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in G_d\}$, where $t(2)=3$, and $t(d)=2$ for $d\geq 3$. The equality of the centralizers can be verified by comparing their dimensions.

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