On the Approximation of the Quantum Gates using Lattices
A. Greene, S. B. Damelin
TL;DR
The paper addresses efficient approximation of gates in $SU(2)$ by finite generator sets using a covering-exponent framework. It constructs an efficient universal set $T$ in $PSU(2)$ via a quaternionic embedding $\Phi$ and a deliberate choice of quaternion factors tied to the number 5, linking gate representations to sums of four squares and enabling a concrete estimate of the covering exponent. It proves $K(T) \le 2$ through spectral/Hecke arguments and presents a conjecture that could further tighten this bound by connecting lattice-angle geometry on $S^3$ to approximation efficiency, with extensions to primes $p \equiv 1\ (\mathrm{mod}\ 4)$. The approach exploits the $SU(2)$--$S^3$ correspondence to translate geometric distribution questions on the 3-sphere into complexity bounds for gate synthesis, offering a pathway for practical universal gate constructions in quantum computing.
Abstract
A central question in Quantum Computing is how matrices in $SU(2)$ can be approximated by products over a small set of generators. A topology will be defined on $SU(2)$ so as to introduce the notion of a covering exponent which compares the length of products required to covering $SU(2)$ with $\varepsilon$ balls against the Haar measure of $\varepsilon$ balls. An efficient universal set over $PSU(2)$ will be constructed using the Pauli matrices, using the metric of the covering exponent. Then, the relationship between $SU(2)$ and $S^3$ will be manipulated to correlate angles between points on $S^3$ to give a conjecture on the maximum of angles between points on a lattice. It will be shown how this conjecture can be used to compute the covering exponent. Some extensions are discussed.