A Simple Derivation of the Fong-Wandzura Pulse Sequence

TL;DR

This paper provides an analytical derivation of the Fong-Wandzura pulse sequence for two-qubit gates in exchange-only spin qubits, simplifying its understanding and construction.

Contribution

It offers a straightforward, pen-and-paper derivation of the Fong-Wandzura sequence, previously known only through numerical methods.

Findings

Analytic derivation of the Fong-Wandzura sequence

Simplification of two-qubit gate construction

Sequence can be derived with basic quantum spin knowledge

Abstract

We give an analytic construction of a class of two-qubit gate pulse sequences that act on five of the six spin-$\frac12$ 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. We thereby analytically derive the Fong-Wandzura sequence that has so far escaped intuitive explanation.