Universality of single qudit gates

Abstract

We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}=\{g_1,\dots ,g_n\}\subset G$ 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 $\mathrm{Ad}:G\to SO(\mathfrak{g})$ 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.

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

References in corpus (3)

• Silica-on-Silicon Waveguide Quantum Circuits
• Classical simulation of fermionic linear optics augmented with noisy ancillas
• Universal extensions of restricted classes of quantum operations

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