Simple derivation of the Fong-Wandzura pulse sequence

Abstract

We give an analytic construction of a class of two-qubit gate pulse sequences that act on five of the six spin-$\frac{1}{2}$ particles used to encode a pair of exchange-only three-spin qubits. Within this class, the problem of gate construction reduces to that of finding a smaller sequence that acts on four spins and is subject to a simple constraint. The optimal sequence satisfying this constraint yields a two-qubit gate sequence equivalent to that found numerically by Fong and Wandzura. Our construction is sufficiently simple that it can be carried out entirely with pen, paper, and knowledge of a few basic facts about quantum spin.