Abstract

We propose and analyze a family of periodic braiding protocols in systems with multiple localized Majorana modes ($\textit{majoranas}$) for the purposes of Hamiltonian engineering. The protocols rely on double braids$-\textit{draids}-$which flip the signs of both majoranas, as one is taken all the way around the other. Rapid draiding dynamically suppresses some or all inter-majorana couplings. Protocols suppressing all couplings can drastically reduce residual dynamics within the nearly degenerate many-body subspace ("majorana purification") producing more robust computational subspace. Non-trivial topological models can be achieved by selectively applying draids to some of overlapping (imperfect) majoranas. Importantly, draids can be implemented without having to physically braid majoranas or using projective measurements. In particular, draids can be performed by periodically modulating the coupling between a quantum dot and topological superconducting wire to dynamically suppress the hybridization of majoranas by more than an order of magnitude in current experimental setups.

Highlights

  • Isolated Majorana fermions play a key role in a number of proposals for topological quantum computing

  • An effective local Hamiltonian description applies for extended times and an appropriate choice of a periodic braiding scheme can be used to engineer a wide variety of effective Hamiltonians. We describe some such schemes applied to a generic starting Hamiltonian operating on a set of localized majorana modes to construct the simple Kitaev chain with localized edge majoranas and the Z2 × Z2 symmetry-protected topological (SPT) insulator; the above is shown to be extended to creating one-dimensional models with an arbitrary number n ∈ Z majorana zero modes

  • The Landau-Zener protocol we propose does not involve physically braiding majoranas, we show that it is (i) exponentially precise in giving a −1 sign to the majorana on the quantum wire, as desired when performing a draid between a constituent majorana in the quantum dot and the majorana on the topological quantum wire, and (ii) any dynamical phase accrued in this process is exponentially small in the distance between the majoranas on the wire

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Summary

INTRODUCTION

Isolated Majorana fermions (or majoranas for short) play a key role in a number of proposals for topological quantum computing. Central to our protocol is the role of double braids, termed draids, which do not change majorana locations, but instead administer a π phase to them Using this sign flip, which is of a robust, topological origin, dynamical decoupling can be performed provided the braiding frequency is faster than the largest overlap between any two majoranas in the system. Our protocol involves performing repeated Landau-Zener tunneling between a quantum dot housing a single fermionic mode, and the superconducting wire, by varying the local potential of the quantum dot in time; see Fig. 8 for illustration We show that this scheme leads to a robust π -phase jump in one of the majorana modes of the quantum wire, which, in particular, can lead to more than a few orders of magnitude suppression of the hybridization between the majoranas in quantum wires in current experiments. We discuss how these draids can be performed in the same setup in a measurement-based scheme

ELEMENTARY OPERATION
SUMMARY OF THE POLYFRACTAL PROTOCOL
HAMILTONIAN CANCELLATION
One-dimensional topological phases
One-dimensional Kitaev phase
Extension to other one-dimensional fermionic topological phases
EXPERIMENTAL IMPLEMENTATION
Analytical argument
MULTIWIRE ARCHITECTURE FOR PERFORMING DRAIDING USING LANDAU-ZENER TRANSITIONS
Close analogy with physical draiding
Architecture for arbitrarily long-range draiding
Measurement-based approach for use in tandem or as an alternative
VIII. DISCUSSION AND CONCLUSIONS
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