Majorana Braiding Research Articles

High-quality thickness-tunable InAs nanowire crosses grown by molecular-beam epitaxy

InAs nanowire crosses show great potential applications in detection and braiding of Majorana zero modes. Controlled growth of high-quality and diameter tunable InAs nanowire crosses is fundamental for these applications. However, it is still difficult to freely and conveniently adjust the diameter of the free-standing InAs nanowire crosses grown by the conventional growth methods. Here, we report a new technique to realize the growth of high-quality thickness-tunable InAs nanowire crosses by molecular-beam epitaxy. GaAs nanowire crosses were firstly grown on the Si (100) substrates spontaneously by merging the <111>-oriented GaAs nanowires. InAs nanowire crosses were then obtained by in situ growth of InAs shells on the facets of GaAs nanowire cross cores. Detailed scanning and transmission electron microscopic observations and energy dispersive spectrum analyses confirm that the InAs nanowire crosses grown by this manner have continuous and smooth morphology and they are high-quality zinc-blende crystals. More importantly, the InAs shell is grown with the vapor-solid growth mechanism and the thickness of the InAs nanowire crosses can be tuned by varying the InAs shell growth time. Our work provides a valuable method for the controlled growth of thickness-tunable semiconductor nanowire crosses.

Two-dimensional superconductors with intrinsic p-wave pairing or nontrivial band topology

Over the past fifteen years, tremendous efforts have been devoted to realizing topological superconductivity in realistic materials and systems, predominately propelled by their promising application potentials in fault-tolerant quantum information processing. In this article, we attempt to give an overview on some of the main developments in this field, focusing in particular on two-dimensional crystalline superconductors that possess either intrinsic p-wave pairing or nontrivial band topology. We first classify the three different conceptual schemes to achieve topological superconductor (TSC), enabled by real-space superconducting proximity effect, reciprocal-space superconducting proximity effect, and intrinsic TSC. Whereas the first scheme has so far been most extensively explored, the subtle difference between the other two remains to be fully substantiated. We then move on to candidate intrinsic or p-wave superconductors, including Sr2RuO4,UTe2,Pb3Bi, and graphene-based systems. For TSC systems that rely on proximity effects, the emphases are mainly on the coexistence of superconductivity and nontrivial band topology, as exemplified by transition metal dichalcogenides, cobalt pnictides, and stanene, all in monolayer or few-layer regime. The review completes with discussions on the three dominant tuning schemes of strain, gating, and ferroelectricity in acquiring one or both essential ingredients of the TSC, and optimizations of such tuning capabilities may prove to be decisive in our drive towards braiding of Majorana zero modes and demonstration of topological qubits.

Josephson effect in topological semimetal-superconductor heterojunctions

Topological semimetals are exotic phases of quantum matter with gapless electronic excitation protected by symmetry. Benefitting from its unique relativistic band dispersion, topological semimetals host abundant quantum states and quantum effects, such as Fermi-arc surface states and chiral anomaly. In recent years, due to the potential application in topological quantum computing, the hybrid system of topology and superconductivity has aroused wide interest in the community. Recent experimental progress of topological semimetal-superconductor heterojunctions is reviewed in two aspects: 1) Josephson current as a mode filter of different topological quantum states; 2) detection and manipulation of topological superconductivity and Majorana zero modes. For the former, utilizing Josephson interference, ballistic transport of Fermi-arc surface states is revealed, higher-order topological phases are discovered, and finite-momentum Cooper pairing and superconducting diode effect are realized. For the latter, by detecting a.c. Josephson effect in Dirac semimetals, the 4π-periodic supercurrent is discovered. By all-electric gate control, the topological transition of superconductivity is obtained. Outlooks of future research on topological semimetal-superconductor heterojunctions and their application in Majorana braiding and topological quantum computing are discussed.

Selective area epitaxy of PbTe-Pb hybrid nanowires on a lattice-matched substrate

Topological quantum computing is based on braiding of Majorana zero modes encoding topological qubits. A promising candidate platform for Majorana zero modes is semiconductor-superconductor hybrid nanowires. The realization of topological qubits and braiding operations requires scalable and disorder-free nanowire networks. Selective area growth of in-plane InAs and InSb nanowires, together with shadow-wall growth of superconductor structures, have demonstrated this scalability by achieving various network structures. However, the noticeable lattice mismatch at the nanowire-substrate interface, acting as a disorder source, imposes a serious obstacle along with this roadmap. Here, combining selective area and shadow-wall growth, we demonstrate the fabrication of PbTe-Pb hybrid nanowires - another potentially promising Majorana system - on a nearly perfectly lattice-matched substrate CdTe, all done in one molecular beam epitaxy chamber. Transmission electron microscopy shows the single-crystal nature of the PbTe nanowire and its atomically sharp and clean interfaces to the CdTe substrate and the Pb overlayer, without noticeable inter-diffusion or strain. The nearly ideal interface condition, together with the strong screening of charge impurities due to the large dielectric constant of PbTe, hold promise towards a clean nanowire system to study Majorana zero modes and topological quantum computing.

Unitary symmetry-protected non-Abelian statistics of Majorana modes

Symmetry-protected topological superconductors (TSCs) can host multiple Majorana zero modes (MZMs) at their edges or vortex cores, while whether the Majorana braiding in such systems is non-Abelian in general remains an open question. Here we uncover in theory the unitary symmetry-protected non-Abelian statisitcs of MZMs and propose the experimental realization. We show that braiding two vortices with each hosting $N$ unitary symmetry-protected MZMs generically reduces to $N$ independent sectors, with each sector braiding two different Majorana modes. This renders the unitary symmetry-protected non-Abelian statistics. As a concrete example, we demonstrate the proposed non-Abelian statistics in a spin-triplet TSC which hosts two MZMs at each vortex and, interestingly, can be precisely mapped to a quantum anomalous Hall insulator. Thus the unitary symmetry-protected non-Abelian statistics can be verified in the latter insulating phase, with the application to realizing various topological quantum gates being studied. Finally, we propose a novel experimental scheme to realize the present study in an optical Raman lattice. Our work opens a new route for Majorana-based topological quantum computation.

Bath-induced decoherence in finite-size Majorana wires at non-zero temperature

Braiding Majorana zero-modes around each other is a promising route toward topological quantum computing. Yet, two competing maxims emerge when implementing Majorana braiding in real systems: on the one hand, perfect braiding should be conducted adiabatically slowly to avoid non-topological errors. On the other hand, braiding must be conducted fast such that decoherence effects introduced by the environment are negligible, which are generally unavoidable in finite-size systems. This competition results in an intermediate time scale for Majorana braiding that is optimal, but generally not error-free. Here, we calculate this intermediate time scale for a T-junction of short one-dimensional topological superconductors coupled to a bosonic bath that generates fluctuations in the local electric potential, which stem from, e.g. environmental photons or phonons of the substrate. We thereby obtain boundaries for the speed of Majorana braiding with a predetermined gate fidelity. Our results emphasize the general susceptibility of Majorana-based information storage in finite-size systems and can serve as a guide for determining the optimal braiding times in future experiments.

Minimal setup for non-Abelian braiding of Majorana zero modes

Braiding Majorana zero modes (MZMs) is the key procedure toward topological quantum computation. However, the complexity of the braiding manipulation hinders its experimental realization. Here we propose an experimental setup consisting of MZMs and a quantum dot state which can substantially simplify the braiding protocol of MZMs. Such braiding scheme, corresponding to a specific closed loop in the parameter space, is quite universal and can be realized in various platforms. Moreover, the braiding results can be measured and manifested through electric current, providing a simple and novel way to detect the non-Abelian statistics of MZMs.

Simulating the dynamics of braiding of Majorana zero modes using an IBM quantum computer

We preform dynamic state evolution on an IBM quantum computer using a simulation of braiding Majorana zero modes as a benchmark for success. We find the native quantum gates introduce too much noise to observe braiding. Instead, we use Qiskit Pulse to develop scaled 2-qubit quantum gates that better match the unitary time evolution operator and enable us to observe braiding. This paper demonstrates that quantum computers can be used for simulation, and highlights the use of pulse-level control for programming quantum computers and constitutes the first experimental evidence of braiding via dynamical Hamiltonian evolution.

Conductance quantization in topological Josephson trijunctions

Here, the authors predict that topological Josephson trijunctions generically display a nonzero quantized transconductance. This result originates from a combination of two different aspects of topology: topological superconductivity in the leads, thus hosting Majorana end modes, and a synthetic topological band structure in the multiterminal Josephson junction. Many groups are currently studying multiterminal junctions using various superconducting hybrid platforms. An experimental check of this prediction would constitute an important milestone on the roadmap toward demonstrating Majorana braiding in trijunctions.

Circulator function in a Josephson junction circuit and braiding of Majorana zero modes

We propose a scheme for the circulator function in a superconducting circuit consisting of a three-Josephson junction loop and a trijunction. In this study we obtain the exact Lagrangian of the system by deriving the effective potential from the fundamental boundary conditions. We subsequently show that we can selectively choose the direction of current flowing through the branches connected at the trijunction, which performs a circulator function. Further, we use this circulator function for a non-Abelian braiding of Majorana zero modes (MZMs). In the branches of the system we introduce pairs of MZMs which interact with each other through the phases of trijunction. The circulator function determines the phases of the trijunction and thus the coupling between the MZMs to gives rise to the braiding operation. We modify the system so that MZMs might be coupled to the external ones to perform qubit operations in a scalable design.

Pumped heat and charge statistics from Majorana braiding

We examine the heat and charge transport of a driven topological superconductor. Our particular system of interest consists of a Y-junction of topological superconducting wires, hosting non-Abelian Majorana zero modes at their edges. The system is contacted to two leads which act as continuous detectors of the system state. We calculate, via a scattering matrix approach, the full counting statistics of the driven heat transport, between two terminals contacted to the system, for small adiabatic driving and characterise the energy transport properties as a function of the system parameters (driving frequency, temperature). We find that the geometric, dynamic contribution to the pumped heat statistics results in a correction to the Gallavotti-Cohen type fluctuation theorem for quantum heat transfer. Notably, the correction term to the fluctuation theorem extends to cycles which correspond to topologically protected braiding of the Majorana zero modes. This geometric correction to the fluctuation theorem differs from its analogs in previously studied systems in that (i) it is non-vanishing for adiabatic cycles of the system's parameters, without the need for cyclic driving of the leads and (ii) it is insensitive to small, slow fluctuations of the driving parameters due to the topological protection of the braiding operation.

Search for non-Abelian Majorana braiding statistics in superconductors

This is a tutorial review of methods to braid non-Abelian anyons (Majorana zero-modes) in topological superconductors. That ``Holy Grail'' of topological quantum information processing has not yet been reached in the laboratory, but there now exists a variety of platforms in which one can search for the Majorana braiding statistics. After an introduction to the basic concepts of braiding we discuss how one might be able to braid immobile Majorana zero-modes, bound to the end points of a nanowire, by performing the exchange in parameter space, rather than in real space. We explain how Coulomb interaction can be used to both control and read out the braiding operation, even though Majorana zero-modes are charge neutral. We ask whether the fusion rule might provide for an easier pathway towards the demonstration of non-Abelian statistics. In the final part we discuss an approach to braiding in real space, rather than parameter space, using vortices injected into a chiral Majorana edge mode as ``flying qubits''.

Vortex Majorana braiding in a finite time

Abrikosov vortices in Fe-based superconductors are a promising platform for hosting Majorana zero modes. Their adiabatic exchange is a key ingredient for Majorana-based quantum computing. However, the adiabatic braiding process can not be realized in state-of-the-art experiments. We propose to replace the infinitely slow, long-path braiding by only slightly moving vortices in a special geometry without actually physically exchanging the Majoranas, like a Majorana carousel. Although the resulting finite-time gate is not topologically protected, it is robust against variations in material specific parameters and in the braiding-speed. We prove this analytically. Our results carry over to Y-junctions of Majorana wires.

Number-conserving analysis of measurement-based braiding with Majorana zero modes

Majorana-based quantum computation seeks to encode information non-locally in pairs of Majorana zero modes, thereby isolating qubit states from a local noisy environment. In addition to long coherence times, the attractiveness of Majorana-based quantum computing relies on achieving topologically protected Clifford gates from braiding operations. Recent works have conjectured that mean-field BCS calculations may fail to account for non-universal corrections to the Majorana braiding operations. Such errors would be detrimental to Majorana-based topological quantum computing schemes. In this work, we develop a particle-number conserving approach for measurement-based topological quantum computing and investigate the effect of quantum phase fluctuations. We demonstrate that braiding transformations are indeed topologically protected in charge-protected Majorana-based quantum computing schemes.

Chiral Majorana fermion

The chiral Majorana fermion, is a massless fermionic particle being its own antiparticle, which was predicted to live in (1+1)D (i.e. one-dimensional space plus one-dimensional time) or (9+1)D. In condensed matter physics, one-dimensional (1D) chiral Majorana fermion can be viewed as the 1/2 of the chiral Dirac fermion, which could arise as the quasiparticle edge state of a two-dimensional (2D) topological state of matter. The appearance of an odd number of 1D chiral Majorana fermions on the edge implies that there exist the non-Abelian defects in the bulk. The chiral Majorana fermion edge state can be used to realize the non-Abelian quantum gate operations on electron states. Starting with the topological states in 2D, we illustrate the general and intimate relation between chiral topological superconductor and quantum anomalous Hall insulator, which leads to the theoretical prediction of the chiral Majorana fermion from the quantum anomalous Hall plateau transition in proximity to a conventional s-wave superconductor. We show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes, and may be used for the topological quantum computation.

Non-abelian statistics of Majorana modes and the applications to topological quantum computation

Since their prediction as fundamental particles in 1937, Majorana fermions have drawn lots of interests in particle physics and dark matter. Their counterparts in condensed matter physics, Majorana zero-Modes (MZMs), have attracted remarkable attention in condensed matter for their potential in building a fault-tolerant quantum computer. Due to the relentless effort, lots of important progress has been made in Majorana physics in the past two decades, as introduced in several excellent review articles. This review focuses on the non-Abelian statistics of MZMs and their application to quantum computation. In the first section of this work, the theoretical progress in searching for MZM is briefly reviewed and the latest experimental progresses are summarized. We next introduce the basic concepts of non-Abelian statistics of MZMs and explain how they can be applied to quantum computation. We then discuss two key experiments to implementing quantum computers in the MZM platform: MZM braiding and MZM qubit readout. In this part, several representative proposals for the Majorana braiding and MZM qubit readout are elaborated. Finally, we introduce a latest concept, the symmetry-protected non-Abelian braiding of Majorana Kramers pairs in time-reversal invariant topological superconductors.

Majorana braiding in realistic nanowire Y-junctions and tuning forks

Majorana fermions are predicted to arise at the ends of nanowire devices which combine superconductivity, strong spin-orbit coupling and an external magnetic field. By manipulating networks of these devices with suitable gating, it has been suggested that braiding operations may be performed which act as logic operations, suitable for quantum computation. However, the unavoidable misalignment of the magnetic field in any realistic device geometry has raised questions about the feasibility of such braiding. In this paper, we numerically simulate braiding operations in devices with Y-junction and tuning fork geometries using an experimentally motivated nanowire model. We study how the static and dynamical features vary with geometric parameters and identify parameter choices that optimise the probability of a successful braid. Notably, we find that there is an optimal Y-junction half-angle (about 20 degrees for our parameter values), which balances two competing mechanisms that reduce the energy gap to excitations. In addition, we find that a tuning fork geometry has significant advantages over a Y-junction geometry, as it substantially reduces the effect of dynamical phase oscillations that complicate the braiding process. Our results suggest that performing a successful braid is in principle possible with such devices, and lies within experimental reach.

Effects of decoherence on diabatic errors in Majorana braiding

The braiding of two non-Abelian Majorana modes is important for realizing topological quantum computation. It can be achieved through tuning the coupling between the two Majorana modes to be exchanged and two ancillary Majorana modes. However, this coupling also makes the braiding subject to environment-induced decoherence. Here, we study the effects of decoherence on the diabatic errors in the braiding process for a set of time-dependent Hamiltonians with finite smoothness. To this end, we employ the master equation to calculate the diabatic excitation population for three kinds of decoherence processes. (1) Only pure dehasing: the scaling of the excitation population changed from $T^{-2k-2}$ to $T^{-1}$ ($k$ is the number of the Hamiltonian's time derivatives vanishing at the initial and final times) as the braiding duration $T$ exceeds a certain value. (2) Only relaxation: the scaling transforms from $T^{-2k-2}$ to $T^{-2}$ for $k=0$ and to $T^{-a}$ ($a>3$) for $k>0$. (3) Pure dephasing and relaxation: the original scaling switches to $T^{-1}$ firstly and then evolves to $T^{-2}$ in the adiabatic limit. Interestingly, the third scaling-varying style holds even when the expectation of pure dephasing rate is much smaller than that of the relaxation rate, which is attributed to the vanishing relaxation at the turning points of the braiding.

Entangling spins in double quantum dots and Majorana bound states

We study the coupling between a singlet-triplet qubit realized in a double quantum dot to a topological qubit realized by spatially well-separated Majorana bound states. We demonstrate that the singlet-triplet qubit can be leveraged for readout of the topological qubit and for supplementing the gate operations that cannot be performed by braiding of Majorana bound states. Furthermore, we extend our setup to a network of singlet-triplet and topological hybrid qubits that paves the way to scalable fault-tolerant quantum computing.

Fractional Josephson Vortices and Braiding of Majorana Zero Modes in Planar Superconductor-Semiconductor Heterostructures.

We consider the one-dimensional (1D) topological superconductor that may form in a planar superconductor-metal-superconductor Josephson junction in which the metal is subjected to spin-orbit coupling and to an in-plane magnetic field. This 1D topological superconductor has been the subject of recent theoretical and experimental attention. We examine the effect of a perpendicular magnetic field and a supercurrent driven across the junction on the position and structure of the Majorana zero modes that are associated with the topological superconductor. In particular, we show that under certain conditions the Josephson vortices fractionalize to half-vortices, each carrying half of the superconducting flux quantum and a single Majorana zero mode. Furthermore, we show that the system allows for a current-controlled braiding of Majorana zero modes.