Abstract
A universal quantum computer requires a full set of basic quantum gates. With Majorana bound states one can form all necessary quantum gates in a topologically protected way, bar one. In this manuscript we present a protocol that achieves the missing, so called, $\pi/8$ 'magic' phase gate. The protocol is based on the manipulation of geometric phases in a universal manner, and does not require fine tuning for distinct physical realizations. The protocol converges exponentially with the number of steps in the geometric path. Furthermore, the magic gate protocol relies on the most basic hardware previously suggested for topologically protected gates, and can be extended to any-phase-gate, where $\pi/8$ is substituted by any $\alpha$.
Highlights
AND MAIN RESULTSIn two landmark papers, Kitaev suggested that nonAbelian anyons could be used to store and process quantum information in a topologically protected way [1,2]
The experimental efforts recently shifted from a mere detection of Majorana signatures to concrete steps towards the realization of platforms that reveal their non-Abelian statistics and allow for quantum information processing via braiding [18,19]
We show in the text that this gives rise to a −π=4 phase gate, meaning a phase ∓π=4 for each fusion channel. (The minus sign appears due to the clockwise orientation of the trajectory and the convention we chose.) Since we can make one of the coupling constants exponentially smaller than the other two, the trajectory in the parameter space is glued to the edges of the octant making the accumulated Berry phase difference equal to −π=2 with exponential accuracy
Summary
Kitaev suggested that nonAbelian anyons could be used to store and process quantum information in a topologically protected way [1,2]. We present a robust scheme for obtaining a Majorana π=8 gate, which is insensitive to such machine control imprecision The protection against such errors arises from universal geometric properties of the Majorana Hilbert space, along with the topological properties of the system. The magic-gate scheme we outline can be realized in any system where the coupling between Majorana states can be controlled, even if imprecisely. The Majorana state at the tip of this coupling is an exact zero mode of the Hamiltonian (i.e., it commutes with the Hamiltonian) This robustness (i.e., Δa 1⁄4 0 if δa 1⁄4 0) lies at the heart of the protected Majorana braiding process and is crucial for the π=8 gate we discuss in this paper
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