Inverse-Closed Dense SU(2) Gate Set

Write $SU(2):=\{M\in M_2(\mathbb{C}):M^{\ast }M=I_2,\det M=1\}$. An inverse-closed dense $SU(2)$ gate package consists of a finite set $G\subset SU(2)$ of exact matrices with finite labels such that $g^{-1}=g^{\ast }$ belongs to $G$ for each $g\in G$ and the subgroup $⟨G⟩$ of finite products is dense in $SU(2)$ in operator norm. A word is a finite ordered list of labels; its value is the corresponding matrix product, and its length is the number of labels.

For a compiler package, also fix a rational $0<{\eta }_0<1\mathrm{/}4096$, a finite list ${\mathcal{N}}_0$ of words, and an integer $L_0$ such that every $U\in SU(2)$ is within operator-norm distance ${\eta }_0\mathrm{/}4$ of the value of some word in ${\mathcal{N}}_0$, and every listed word has length at most $L_0$. This is a finite base-net certificate. Density proves that such a certificate exists, but does not compute the list or its maximum word length.