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arXiv:2004.11385 · PRX Quantum

Double Braiding Majoranas for Quantum Computing and Hamiltonian Engineering

Ivar Martin, Kartiek Agarwal

Published Dec 21, 2020• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.1.020324

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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.
We study a series of dynamical protocols which involve periodically driving a quantum dot coupled to a putative nanowire hosting Majorana zero modes (MZMs) to (i) reduce the hybridization between MZMs, (ii) improve the coherence of the Majorana qubit with respect to 1/f1/f1/f dephasing noise and quasiparticle poisoning, and (iii) provide a definitive test to differentiate Andreev bound states (ABSs) from MZMs. The protocols are based on the notion of draiding, exchanging a pair of Majoranas twice, repeatedly, at high frequency [I. Martin and K. Agarwal, PRX Quantum 1, 020324 (2020)]. In this process, the exchanged Majorana operators acquire a robust minus sign such that terms in the Hamiltonian, linear in either operator, vanish on average. The four protocols proposed implement draiding by coupling quantum dot(s) to the end(s) of the nanowire. They are treated using Floquet theory and numerical simulations. The hybridization energy and decoherence rate are shown to be reduced by several orders of magnitude, in accordance with theoretical expectations, when the protocols are implemented on nanowires described by experimentally relevant parameters. The tunneling conductance computed in this Floquet setting reveals zero-bias peaks (ZBPs) that become more centered at zero voltage bias. When these protocols are implemented on nanowires supporting ABSs that mimic MZMs in ZBP measurements, the qubit coherence deteriorates, in stark contrast to the case where the nanowire supports MZMs and the coherence drastically improves, thus serving as a dynamical test to distinguish MZMs from trivial bound states.
Majorana-based topological qubits are expected to exploit the nonabelian braiding statistics of Majorana modes in topological superconductors to realize fault-tolerant topological quantum computation.Scalable qubit designs require several Majorana modes localized on quantum wires networks, with braiding operations relying on the presence of the groundstate degeneracy of the topologically nontrivial superconducting phase.However, this degeneracy is lifted due to the hybridization between Majorana modes localized at a finite distance.Here, we describe a braiding protocol in a trijunction where each branch consists of a lattice of Majorana modes overlapping at a finite distance.We find that the energy splitting between the groundstate and the lowest-energy state decreases exponentially with the number of Majorana modes if the system is in its topologically nontrivial regime.This result does not rely on the specific braiding geometry and on the details of the braiding scheme but is a consequence of the supersymmetry and nontrivial topology of the effective low-energy Hamiltonian describing the Majorana lattice.
Majorana zero modes have emerged as one of the most promising platforms for topological quantum computation, since their non-Abelian braiding statistics allow quantum information to be encoded nonlocally and manipulated through braiding operations that are, in principle, protected against local perturbations. In practice, however, a braid is only as robust as its physical implementation: finite-time operation, residual couplings, and environmental noise can all convert local excitations into logical errors during the exchange process. Here, we address this question through a microscopic comparison of two representative braiding architectures, a nanowire trijunction and a quantum-dot-assisted setup, simulating the full time-dependent Bogoliubov--de Gennes dynamics under both noiseless and noisy conditions. We show that the dot-assisted architecture consistently achieves a lower error over a shorter timescale than the trijunction, owing to its more localized exchange mechanism. This advantage persists in the presence of noise, and a spatially resolved analysis further reveals that, in the dot-assisted geometry, fast noise localized on the dot produces a smaller error than equivalent noise on the wires, whereas slow, quasi-static noise on the dot becomes the dominant limitation. Taken together, these findings link the different error contributions directly to device geometry, pointing to concrete design principles for noise-resilient Majorana-based quantum gates.
Topology-related ideas might lead to noise-resilient quantum computing. For example, it is expected that the slow spatial exchange (``braiding'') of Majorana zero modes in superconductors yields quantum gates that are robust against disorder. Here, we report our numerical experiments, which describe the dynamics of a Majorana qubit built from quantum dots controlled by time-dependent gate voltages. Our protocol incorporates nonprotected control, braiding-based protected control, and readout of the Majorana qubit. We use the Kitaev chain model for the simulations, and we focus on the case when the main source of errors is quasistatic charge noise affecting the hybridization energy splitting of the Majorana modes. We provide quantitative guidelines to suppress both diabatic errors and disorder-induced qubit dephasing, such that a fidelity plateau is observed as the hallmark of the topological quantum gate. Our simulations predict realistic features that are expected to be seen in future braiding experiments with Majorana zero modes and other topological qubit architectures.
The realization and manipulation of Majorana zero modes have drawn significant attention for their crucial role in enabling topological quantum computation. Conventional approaches to the braiding of Majorana zero modes rely on adiabatic processes. In this work, using a composite 2-Kitaev-chain system accommodating Majorana zero modes as a working example, we propose a nonadiabatic and non-Abelian geometry phase-based protocol to execute operations on these Majorana zero modes. This is possible by locally coupling the edge sites of both quantum chains with an embedded lattice defect, successfully simulating the braiding operation of two Majorana modes in a highly nonadiabatic fashion. To further enhance the robustness against control imperfections, we apply a multiple-pulse composite strategy to our quantum chain setting for second-order protection of the braiding operations. Our proposal can also support the fast and robust realization of the $\ensuremath{\pi}/8$ gate, an essential ingredient for universal quantum computation. This work hence offers a potential pathway toward the nonadiabatic and fault-tolerant control of Majorana zero modes.
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.