On the Approximation of the Quantum Gates using Lattices

TL;DR

This paper investigates how to efficiently approximate quantum gates in SU(2) using lattice-based methods, introducing a covering exponent to measure the approximation efficiency and constructing universal sets with the Pauli matrices.

Contribution

It defines a new topology on SU(2), introduces the covering exponent concept, constructs a universal set using Pauli matrices, and relates the geometry of SU(2) and S^3 to improve gate approximation methods.

Findings

Defined a topology on SU(2) for approximation analysis

Constructed a universal set over PSU(2) with Pauli matrices

Proposed a conjecture relating lattice angles to approximation efficiency

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.