Spin swap gate in the presence of qubit inhomogeneity in a double quantum dot

We study theoretically the effects of qubit inhomogeneity on the quantum logic gate of qubit swap, which is an integral part of the operations of a quantum computer. Our focus here is to construct a robust pulse sequence for swap operation in the simultaneous presence of Zeeman inhomogeneity for quantum dot trapped electron spins and the finite-time ramp-up of exchange coupling in a double dot. We first present a geometric explanation of spin swap operation, mapping the two-qubit operation onto a single-qubit rotation. We then show that in this geometric picture a square-pulse-sequence can be easily designed to perform swap in the presence of Zeeman inhomogeneity. Finally, we investigate how finite ramp-up times for the exchange coupling $J$ negatively affect the performance of the swap gate sequence, and show how to correct the problems numerically.