On a Problem of Construction of Efficient Universal Sets of Quantum Gates: A preliminary report

Abstract

A 1-qubit gate is defined as an element of the unitary group $U(2)$, the group of complex valued matrices whose conjugate transposes equal their inverses. In quantum computing, it is important to be able to find good approximations of 1-qubit gates by using a dense group generated by a universal subset of the projective special unitary group $G=PSU(2)$. Here, $PSU(2)$ is the group formed by taking all elements of $U(2)$ with determinant 1 and taking the quotient of this subgroup by the group $\{I,-I\}$ where $I$ is the identity matrix. In http://publications.ias.edu/sarnak/paper/2637 [1], Sarnak recently studied this problem motivated by developing a continued fraction type algorithm for this problem. The continued fraction algorithm is the fastest known algorithm to approximate real numbers by the dense subset of rational numbers. In this report, we study the efficiency of the approximation of $G$ by using a dense group generated by a universal subset of $G$. The measure of the efficiency $K(g)$ of an arbitrary universal subset $g$ is defined in [1] and satisfies $K(g)\geqslant 1$. $g$ is more efficient when $K(g)$ is closer to $1$. We address some open problems posed by Sarnak in [1]. This report forms the basis of the shorter preprint [LD].