Universality of single qudit gates
| Autor: | Sawicki, Adam, Karnas, Katarzyna |
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| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | Ann. Henri Poincar\'e, Volume 18, Issue 11, pp 3515-3552, 2017 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00023-017-0604-z |
| Popis: | We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}=\{g_1,\ldots,g_n\}\subset G$ is universal, i.e if the closure $\overline{<\mathcal{S}>}$ is equal to $G$, where $G$ is either the special unitary or the special orthogonal group. To every gate $g$ in $\mathcal{S}$ we asign its image under the adjoint representation $\mathrm{Ad}_g$, where $\mathrm{Ad}:G\rightarrow SO(\mathfrak{g})$ and $\mathfrak{g}$ is the Lie algebra of $G$. The necessary condition for the universality of $\mathcal{S}$ is that the only matrices that commute with all $\mathrm{Ad}_{g_i}$'s are proportional to the identity. If in addition there is an element in $<\mathcal{S}>$ whose Hilbert-Schmidt distance from the centre of $G$ belongs to $]0,\frac{1}{\sqrt{2}}]$, then $\mathcal{S}$ is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of $d$-dimensional gates in a finite number of steps and formulate the general classification theorem. Comment: Significantly improved universality criteria and presentation. A simple algorithm that allows deciding the universality of any set of gates in a finite number of steps added and discussed. Accepted in AHP |
| Databáze: | arXiv |
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