Manipulation of Majorana states in X-junction geometries

TL;DR

This paper explores quantum manipulation using four Majorana bound states in X-junctions, expanding the parameter space and proposing protocols for quantum computation, including non-topologically protected but feasible operations.

Contribution

It introduces the SO(4) symmetry-based X-junction geometry for Majorana manipulation and develops protocols for specific Berry's phase values relevant to quantum computing.

Findings

Derived conditions for maintaining Majorana degeneracy.

Computed Berry's phase for manipulation protocols.

Proposed feasible quantum computation schemes.

Abstract

We study quantum manipulation based on four Majorana bound states in X-junction geometry. The parameter space of this setup is bigger than of the previously studied Y-junction and is described by SO(4) symmetry group. In order for quantum computation to be dephasing free, two Majorana states have to stay degenerate at all times. We find a condition necessary for that and compute the Berry's phase, $2\alpha$, accumulated during the manipulation. We construct simple protocols for the variety of values of $\alpha$, including $\pi/8$ needed for the purposes of quantum computation. Although the manipulations in general X-junction geometry are not topologically protected, they may prove to be a feasible compromise for aims of quantum computation.