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arXiv:1507.02881 · Physical review. B./Physical review. B

Braiding properties of Majorana Kramers pairs

Konrad Wölms, Ady Stern, Karsten Flensberg

Published Jan 19, 2016• 27 citations
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Abstract

We consider the braiding of Kramers pairs of Majorana bound states. We derive the most general transformation on the many-body ground state that is applied as the result of such a braiding process. The result is derived in the context of a toy model, but we will show that it has the most general form that is compatible with local and global conservation of electron parity. In accordance with earlier work, the resulting transformation turns out to be path dependent, which shows that Kramers pairs of Majorana bound states cannot be used for topological quantum computation. We also discuss under which conditions the result is path independent and corresponds to two independent exchanges of pairs of Majorana bound states.

DOI
10.1103/physrevb.93.045417

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We consider the braiding of Kramers pairs of Majorana bound states. We derive the most general transformation on the many-body ground state that is applied as the result of such a braiding process. The result is derived in the context of a toy model, but we will show that it has the most general form that is compatible with local and global conservation of electron parity. In accordance with earlier work, the resulting transformation turns out to be path dependent, which shows that Kramers pairs of Majorana bound states cannot be used for topological quantum computation. We also discuss under which conditions the result is path independent and corresponds to two independent exchanges of pairs of Majorana bound states.
We consider Kramers pairs of Majorana bound states under adiabatic time evolution. This is important for the prospects of using such bound states as parity qubits. We show that local adiabatic perturbations can cause a rotation in the space spanned by the Kramers pair. Hence, the quantum information is unprotected against local perturbations, in contrast to the case of single localized Majorana bound states in systems with broken time reversal symmetry. We give an analytical and a numerical example for such a rotation, and specify sufficient conditions under which a rotation is avoided. We give a general scheme for determining when these conditions are satisfied and exemplify it with a general model of a quasi-1D time reversal symmetric topological superconductor.
Demonstration of a nontrivial result of quasiparticle exchange (or braiding) is usually considered the definitive proof of a topological phase with non-Abelian excitations, such as Majorana bound states (MBSs). However, in finite systems with disorder and smooth potential variations, the MBSs are imperfect in the sense that they are not fully isolated in space and can, to a varying degree, resemble conventional fermions. The main goal of this work is to reveal the braiding properties of isolated MBSs, regular fermions, and anything in between. We start from an experimentally attractive braiding protocol and find a way to compensate for the undesired splitting of the ground-state degeneracy which occurs during the protocol for imperfect MBSs. This leads to a braiding outcome that depends on the degree of MBS isolation but remains robust and non-Abelian except in the perfect fermion limit. Our protocol could be implemented in different platforms with non-Abelian excitations, including quantum-dot-based minimal Kitaev chains.
Qubits built out of Majorana zero modes constitute the primary path toward topologically protected quantum computing. Simulating the braiding process of multiple Majorana zero modes corresponds to the quantum dynamics of a superconducting many-body system. It is crucial to study the Majorana dynamics both in the presence of all other quasiparticles and for reasonably large system sizes. We present a method to calculate arbitrary many-body wave functions as well as their expectation values, correlators, and overlaps from time evolved single-particle states of a superconductor, allowing for significantly larger system sizes. We calculate the fidelity, transition probabilities, and joint parities of Majorana pairs to track the quality of the braiding process. We show how the braiding success depends on the speed of the braid. Moreover, we demonstrate the topological CNOT two-qubit gate as an example of two-qubit entanglement. Our Letter opens the path to test and analyze the many theoretical implementations of Majorana qubits. Moreover, this method can be used to study the dynamics of any noninteracting superconductor.
We develop a complete theory for symmetry protected non-Abelian statistics of Majorana Kramers' pairs (MKPs) in time-reversal (TR) invariant topological superconductors, with fundamental results being uncovered. By introducing an effective Hamiltonian approach to describe the braiding of MKPs, we show that the non-Abelian braiding is protected when the effective Hamiltonian exhibits a new TR-like antiunitary symmetry, which is satisfied if the system is free of dynamical noise. Importantly, even the dynamical noise cannot cause error in braiding, unless the noise correlation function breaks a dynamical TR symmetry. This is a profound result and generalizes the TR symmetry protection of MKPs to the dynamical regime. Moreover, the resulted error by noise is shown to be a higher-order effect, compared with the decoherence of Majorana qubits without TR symmetry protection. This study completes the theory of symmetry-protected non-Abelian statistics of MKPs, and shows that the non-Abelian braiding of MKPs is well observable and may have versatile applications to future quantum computation technologies.
Avenues of Majorana bound states (MBSs) have become one of the primary directions towards a possible realization of topological quantum computation. For a Y junction of Kitaev quantum wires, we numerically investigate the braiding of MBSs while considering the full quasiparticle background. The two central sources of braiding errors are found to be the fidelity loss due to the incomplete adiabaticity of the braiding operation as well as the finite hybridization of the MBSs. The explicit extraction of the braiding phase from the full many-particle states allows us to analyze the breakdown of the independent-particle picture of Majorana braiding. Furthermore, we find nearest-neighbor interactions to significantly affect the braiding performance for better or worse, depending on the sign and magnitude of the coupling.
We study the decoherence dynamics of Majorana qubit braiding operations in a T-junction topological superconducting chain (TSC) system, in which the braiding is performed by controlling the electron-chemical potentials of the TSCs and the couplings between them. By solving rigorously the Majorana qubit dynamics, we show how the Majorana qubit coherence is generated through Bogoliubon correlations formed by exchanging Majorana zero modes (MZMs) between two TSCs in braiding operations. We also demonstrate with the exact master equation how MZMs and also the Bogoliubon correlations dissipate due to charge fluctuations of the controlling gates at both the zero and finite temperatures. As a result, Majorana qubit coherence and also the fermion parity conservation cannot be immune from local perturbations during braiding operations.