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arXiv:1302.1824 · Physical Review Letters

Braiding of Atomic Majorana Fermions in Wire Networks and Implementation of the Deutsch-Jozsa Algorithm

Christina V. Kraus, P. Zoller, Mikhail A. Baranov

Published Nov 11, 2013• 61 citations
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Abstract

We propose an efficient protocol for braiding Majorana fermions realized as edge states in atomic wire networks, and demonstrate its robustness against experimentally relevant errors. The braiding of two Majorana fermions located on one side of two adjacent wires requires only a few local operations on this side which can be implemented using local site addressing available in current experiments with cold atoms and molecules. Based on this protocol we provide an experimentally feasible implementation of the Deutsch-Jozsa algorithm for two qubits in a topologically protected way.

DOI
10.1103/physrevlett.111.203001

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We propose an efficient protocol for braiding Majorana fermions realized as edge states in atomic wire networks, and demonstrate its robustness against experimentally relevant errors. The braiding of two Majorana fermions located on one side of two adjacent wires requires only a few local operations on this side which can be implemented using local site addressing available in current experiments with cold atoms and molecules. Based on this protocol we provide an experimentally feasible implementation of the Deutsch-Jozsa algorithm for two qubits in a topologically protected way.
In this paper, we present a hybrid scheme for topological quantum computation in a system of cold atoms trapped in an atomic lattice. A topological qubit subspace is defined using Majorana fermions which emerge in a network of atomic Kitaev one-dimensional wires. We show how braiding can be efficiently implemented in this setup and propose a direct way to demonstrate the non-Abelian nature of Majorana fermions via a single-parity measurement. We then introduce a proposal for the efficient, robust, and reversible mapping of the topological qubits to a conventional qubit stored in a single atom. There, well-controlled standard techniques can be used to implement the missing gates required for universal computation. Our setup is complemented with an efficient nondestructive protocol to check for errors in the mapping.
We propose and analyze a family of periodic braiding protocols in systems with multiple localized Majorana modes (majoranas) for the purposes of Hamiltonian engineering.The protocols rely on double braids-draids-which flip the signs of both majoranas, as one is taken all the way around the other.Rapid draiding can be used to dynamically suppress some or all intermajorana couplings.Suppressing all couplings can drastically reduce residual majorana dynamics, producing a more robust computational subspace.Nontrivial topological models can be obtained by selectively applying draids to some of the overlapping (imperfect) majoranas.Remarkably, draids can be implemented without having to physically braid majoranas or performing projective measurements.For instance, we show that draids can be performed by periodically modulating the coupling between a quantum dot and a topological superconducting wire to dynamically suppress the hybridization of majoranas in the quantum wire.In current experimental setups, this could lead to suppression of this coupling by a few orders of magnitude.The robustness of this protocol can be shown to parallel the topological robustness of physically braided majoranas.We propose an architecture that implements draids between distant majorana modes within a quantum register using a setup with multiple quantum dots and also discuss measurement-based ways of implementing the same.
Majorana zero modes are a promising platform for topologically protected quantum information processing. Their non-Abelian nature, which is key for performing quantum gates, is most prominently exhibited through braiding. While originally formulated for two-dimensional systems, it has been shown that braiding can also be realized using one-dimensional wires by forming an essentially two-dimensional network. Here, we show that in driven systems far from equilibrium, one can do away with the second spatial dimension altogether by instead using quasienergy as the second dimension. To realize this, we use a Floquet topological superconductor which can exhibit Majorana modes at two special eigenvalues of the evolution operator, 0 and $\ensuremath{\pi}$, and thus can realize four Majorana modes in a single, driven quantum wire. We describe and numerically evaluate a protocol that realizes a topologically protected exchange of two Majorana zero modes in a single wire by adiabatically modulating the Floquet drive and using the $\ensuremath{\pi}$ modes as auxiliary degrees of freedom.
We analyze the control of Majorana zero-energy states by mapping the fermionic system onto a chain of Ising spins. Although the topological protection is lost for the Ising system, the mapping provides additional insight into the nature of the quantum states. By controlling the local magnetic field, one can separate the Ising chain into ferromagnetic and paramagnetic phases, corresponding to topological and nontopological sections of the fermionic system. In this paper we propose (topologically nonprotected) protocols performing the braiding operation, and in fact also more general rotations. We first consider a TTT-junction geometry, but we also propose a protocol for a purely one-dimensional (1D) system. Both setups rely on an extra spin-12\frac{1}{2}21​ coupler. By including the extra spin in the TTT-junction geometry, we overcome limitations due to the 1D character of the Jordan--Wigner transformation. In the 1D geometry the coupler, which controls one of the Ising links, should be manipulated once the ferromagnetic (topological) section of the chain is moved far away. We also propose experimental implementations of our scheme. One is based on a chain of flux qubits which allows for all needed control fields. We also describe how to translate our scheme for the 1D setup to a chain of superconducting wires hosting each a pair of Majorana edge states.
Majorana zero modes are a promising platform for topologically protected quantum information processing. Their non-Abelian nature, which is key for performing quantum gates, is most prominently exhibited through braiding. While originally formulated for two-dimensional systems, it has been shown that braiding can also be realized using one-dimensional wires by forming an essentially two-dimensional network. Here, we show that in driven systems far from equilibrium, one can do away with the second spatial dimension altogether by instead using quasienergy as the second dimension. To realize this, we use a Floquet topological superconductor which can exhibit Majorana modes at two special eigenvalues of the evolution operator, 0 and π, and thus can realize four Majorana modes in a single, driven quantum wire. We describe and numerically evaluate a protocol that realizes a topologically protected exchange of two Majorana zero modes in a single wire by adiabatically modulating the Floquet drive and using the π modes as auxiliary degrees of freedom.
Constructing Majorana zero modes (MZMs) is one of the most promising schemes for achieving topological quantum computing. However, recent research primarily focuses on preparing stable materials that carry MZMs and detecting their existence. Further combining structures that carry MZMs for quantum computing would facilitate the synergistic development of material fabrication and quantum computing. Here, we take the T-junction structure carrying MZMs as the basic unit to construct a two-dimensional (2D) grid structure. Based on this structure, the protocols for performing braiding operations for single-qubit and two-qubit gates are defined. Furthermore, the Deutsch-Jozsa (D-J) algorithm is derived to demonstrate the vast application potential of the 2D grid structure in more complex quantum algorithms.