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 varepsilon > 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 varepsilon > 0 small enough, we define t_varepsilon as the minimum reduced word length such that every point of SU(2) lies within a ball of radius varepsilon 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 δ in \[0, frac{2}{3}\]. Choose f: \(0, infty\) → \(1, infty\) satisfying lim_{varepsilon→ 0+}dfrac{log\(f(t_varepsilon\))}{t_varepsilon} exists with value 0. Assume that the inequality varepsilon leqslant f\(t_varepsilon\)cdot 5^{frac{-t_varepsilon}{6-3δ}} holds. Then K(T) leqslant 2-δ. Our conjecture implies the following: Let ν\(5^t_varepsilon\) denote the set of integer solutions of the quadratic form: x₁²+x₂²+x₃²+x₄²=5^t_varepsilon. Let Mequiv M_S³\(mathcal{N}\) denote the covering radius of the points mathcal{N}=ν\(5^t_varepsilon\)cupν\(5^{t_varepsilon-1}\) on the sphere S³ in mathbb{R}⁴. Then M sim f\(log N\)N^{frac{-1}{6-3δ}}. Here Nequiv N\(varepsilon\)=6cdot5^t_varepsilon-2.