A Note on a Quantitative Form of the Solovay-Kitaev Theorem

The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of $SU(2)$ to approximate an arbitrary element of $SU(2)$. The Solovay-Kitaev Theorem is a well-known theorem that guarantees the existence of a finite sequence of 1-qubit quantum gates approximating an arbitrary unitary matrix in $SU(2)$ within specified accuracy $\epsilon >0$. In this note we study a quantitative description of this theorem in the following sense. We will work with a universal gate set $T$, a subset of $SU(2)$ such that the group generated by the elements of $T$ is dense in $SU(2)$. For $\epsilon >0$ small enough, we define $t_{\epsilon }$ as the minimum reduced word length such that every point of $SU(2)$ lies within a ball of radius $\epsilon$ centered at the points in the dense subgroup generated by $T$. For a measure of efficiency on T, which we denote $K(T)$, we prove the following theorem: Fix a $\delta$ in $[0,\frac{2}{3}]$. Choose $f:(0,\mathrm{\infty })\to (1,\mathrm{\infty })$ satisfying $\lim_{\varepsilon\to 0+}\dfrac{\log(f(t_{\varepsilon}))}{t_{\varepsilon}}$existswithvalue$0$. Assume that the inequality $\varepsilon \leqslant f(t_{\varepsilon})\cdot 5^{\frac{-t_{\varepsilon}}{6-3\delta}}$holds\mathrm{.}Then$K(T) \leqslant 2-\delta$. Our conjecture implies the following: Let $\nu (5^{t_{\epsilon }})$ denote the set of integer solutions of the quadratic form: $x_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\epsilon }}$. Let $M\equiv M_{S^3}($\mathcal{N}$)$denotethecoveringradiusofthepoints$$\mathcal{N}$=\nu(5^{t_{\varepsilon}})\cup\nu(5^{t_{\varepsilon}-1})$ on the sphere $S^3$ in ${\mathbb{R}}^4$. Then $M \sim f(\log N)N^{\frac{-1}{6-3\delta}}$\mathrm{.}Here$N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2$.