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SIGNATURE OF MAJORANA MODES AND ASPECTS OF THEIR BRAIDING

Amit Nag

Published Jan 1, 2018• 0 citations
Publisher

Abstract

Majorana zero modes are emergent zero-energy quasiparticle excitations in certain superconducting systems that can be viewed as fractionalized or “half” electrons. These quasiparticles obey non-Abelian braiding statistics which is one manifestation of such half-electron character. Due to such non-Abelian braiding property, Majorana zero mode pairs hold promise as potential qubits for topological quantum computation. It is somewhat surprising that, at least theoretically, ordinary one-dimensional semiconductor systems can be induced to host such esoteric Majorana modes as edge states if some precise experimental conditions are satisfied. Because of the relative simplicity of material and experimental requirements to host Majorana modes, there has been a flurry of experimental effort to realize them in semiconductor nanowire systems. While experimental efforts have produced preliminary evidence for the presence of Majorana zero modes in these systems, a thorough confirmation is lacking. The experimental signature in question is the presence of a zero-bias conductance-peak that, while necessary, is not a sufficient criterion to establish the presence of underlying Majorana modes. Given the importance of Majorana braiding for topological quantum computation and skepticism over the presence of Majorana modes in these experimental systems, it would seem natural to attempt braiding these putative Majorana modes in the near future. In that case, an observation of non-Abelian statistics would provide the necessary and sufficient condition in favor of Majorana presence in the studied experimental systems. This thesis has three distinct parts. First, we assume perfect Majorana modes as given that can be successfully braided. In this case, we calculate the diabatic error due to the finite speed of braiding when the system is coupled to a Bosonic bath. Next, we grant that the mechanism for zero-bias conductance-peak is indeed topological, albeit the underlying Majorana modes may be imperfect (the modes are not precisely at zero energy). We study the interplay of dissipation and finite energy splitting of the Majorana modes and study its consequence regarding the probability of successful braiding. Lastly, we propose studying the correlation between independent left and right conductance measurements as a means to distinguish between a topological versus a non-topological the mechanism underlying the observed zero-bias conductance-peak.

DOI
10.13016/za9o-pibw

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Majorana zero modes are emergent zero-energy quasiparticle excitations in certain superconducting systems that can be viewed as fractionalized or “half” electrons. These quasiparticles obey non-Abelian braiding statistics which is one manifestation of such half-electron character. Due to such non-Abelian braiding property, Majorana zero mode pairs hold promise as potential qubits for topological quantum computation. It is somewhat surprising that, at least theoretically, ordinary one-dimensional semiconductor systems can be induced to host such esoteric Majorana modes as edge states if some precise experimental conditions are satisfied. Because of the relative simplicity of material and experimental requirements to host Majorana modes, there has been a flurry of experimental effort to realize them in semiconductor nanowire systems. While experimental efforts have produced preliminary evidence for the presence of Majorana zero modes in these systems, a thorough confirmation is lacking. The experimental signature in question is the presence of a zero-bias conductance-peak that, while necessary, is not a sufficient criterion to establish the presence of underlying Majorana modes. Given the importance of Majorana braiding for topological quantum computation and skepticism over the presence of Majorana modes in these experimental systems, it would seem natural to attempt braiding these putative Majorana modes in the near future. In that case, an observation of non-Abelian statistics would provide the necessary and sufficient condition in favor of Majorana presence in the studied experimental systems. This thesis has three distinct parts. First, we assume perfect Majorana modes as given that can be successfully braided. In this case, we calculate the diabatic error due to the finite speed of braiding when the system is coupled to a Bosonic bath. Next, we grant that the mechanism for zero-bias conductance-peak is indeed topological, albeit the underlying Majorana modes may be imperfect (the modes are not precisely at zero energy). We study the interplay of dissipation and finite energy splitting of the Majorana modes and study its consequence regarding the probability of successful braiding. Lastly, we propose studying the correlation between independent left and right conductance measurements as a means to distinguish between a topological versus a non-topological the mechanism underlying the observed zero-bias conductance-peak.
Braiding Majorana zero modes (MZMs) is the key procedure toward topological quantum computation. We show such braiding can be well performed in a parallel semiconductor-superconductor nanowire structure. Considering the fact that the low-energy Andreev bound states (ABSs) usually mix with the MZMs in the present setup, we further investigate the braiding properties of MZMs when an ABS is presented. Our numerical simulation suggests that ABS can be regarded as a pair of weakly coupled MZMs. The dynamical hybridization of MZMs plus the non-Abelian braiding of MZMs would induce an arbitrary rotation on the Bloch sphere of a single qubit. Remarkably, such rotation is manipulable since the rotation parameters could be individually modulated. Thus, the dynamic evolution can be eliminated and the non-Abelian braiding statistics, independent of the braiding time, retrieves.
Majorana zero modes, characterized by non-Abelian exchange statistics, have been widely viewed as prime candidates for implementing topological quantum computing. The associated braiding and fusion operations realize topologically protected quantum gates and qubit measurements, thus providing a feasible pathway toward intrinsically fault-tolerant quantum computation. However, the experimental realization of controllable braiding and fusion of Majorana zero modes remains one of the most formidable challenges in the field.In this paper, building upon recent theoretical advances in quantum-dot-assisted schemes for realizing the braiding and fusion of Majorana zero modes, we provide a systematic introduction to the underlying theoretical foundations and detailed operational procedures, and further elucidate the experimental feasibility of the approach.Furthermore, this paper presents two lines of investigation developed within the proposed framework. Firstly, based on the quantum-dot-assisted braiding scheme, the dynamical evolution of Andreev bound states in Majorana nanowires is revealed. Secondly, employing the quantum-dot-assisted fusion scheme, numerical simulations of charge transfer along different fusion paths clarify the nontrivial fusion rules of Majorana zero modes and the corresponding operational procedures.
Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its susceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topological physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic error sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement-based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poisoning, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters.While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic environment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1/f1/f1/f noise, thermal quasiparticle excitations, and phonons for different qubit sizes.The residual errors discussed above will necessarily require error correction for a sufficiently long quantum computation. We develop physically motivated noise models for Majorana-based qubits that can be used to analyze the performance of a quantum error correcting code. We apply this noise model to estimate pseudo-thresholds for a small subsystem code, identifying the relative importance of difference error processes from a fault tolerance perspective. Our results emphasize the necessity of suppressing long-lived quasiparticle excitations that can spread across the code.Finally, we turn our attention to a different platform that could host non-Abelian topological defects: fractional Chern insulators in graphene. We study the edge states of fractional Chern insulators using the field theory of fractional quantum Hall edges supplemented with a symmetry action. We find that lattice symmetries impose a quantized momentum difference for edge electrons in a fractional state of a C=2C=2C=2 Chern band. This momentum difference can be used to selectively contact the different edge states, thereby allowing detection of topological defects in the bulk with a standard four terminal measurement. Our proposal could be implemented in graphene subject to an artificially patterned lattice.
Majorana zero modes (MZMs) localized in superconducting vortices are predicted to realize Ising anyon non-Abelian statistics. In finite-size vortex lattices, overlapping MZM wavefunctions generate exponentially decaying energy splittings. We investigate whether these finite-size effects merely perturb the ideal Ising braid group representation or fundamentally alter it into a distinct, dynamically dependent representation. By modeling a two-dimensional topological superconductor with an effective low-energy Hamiltonian incorporating exponentially decaying overlap terms, we numerically evaluate the time-evolution operator during adiabatic braiding. Our analysis shows that finite-size effects introduce a non-commuting dynamical phase that modifies the ideal Ising braid matrix. This results in a time-dependent braid matrix that alters expected interferometric conductance oscillations. Specifically, the conductance oscillations exhibit a phase shift proportional to the braiding time and inter-vortex distance, scaling as δϕ∝τΔe−R/ξ\delta \phi \propto \tau \Delta e^{-R/\xi}δϕ∝τΔe−R/ξ [to verify]. The non-Abelian nature appears degraded, with the braid matrix depending on the adiabatic trajectory and duration rather than solely on its topological class. This suggests interferometry can dynamically detect these finite-size modifications. ## 1. Introduction Majorana zero modes (MZMs) in topological superconductors are predicted to exhibit non-Abelian braiding statistics, forming the basis for topological quantum computation. In a p+ipp+ipp+ip superconductor or similar platforms, vortices trap MZMs whose braiding is described by the Ising anyon model. However, realistic systems are finite, and MZMs localized at neighboring vortices have overlapping wavefunctions, leading to exponentially small energy splittings. While often treated as a negligible perturbation, the impact of these splittings on the braid group representation during finite-time braiding remains an open question. Here, we analyze the
Abstract 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.
Braiding Majorana zero-modes around each other is a promising route towards topological quantum computing. Yet, two competing maximes 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 unavoidable decoherence effects introduced by the environment are negligible. 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.