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Transport of Majorana zero modes in topological superconducting wires

Bill P. Truong

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Majorana zero modes (MZMs) appear as edge states in topological superconductors and are envisioned as promising candidates for application towards fault-tolerant quantum computation. Quantum information may be stored in a degenerate ground state subspace constructed from MZMs while quantum gates may be performed by braiding MZMs and exploiting their non-Abelian properties. While the topological protection of MZMs ensures that they are robust against decoherence, their dynamical manipulation can nonetheless induce undesirable transitions to excited states, manifesting as a “diabatic error.”This thesis studies the diabatic error accumulated as MZMs are transported across a superconducting wire, which is an essential building block for physical braiding schemes. The transport protocol considered relies on appropriately tuning sections (“piano keys”) of a wire between distinct topological phases via electric gates to control local chemical potentials. First, the diabatic error is examined for transport which uses multiple piano keys. The error is calculated through numerical simulation of the transport protocol and is complemented by approximate analytical expressions calculated using an effective, low-energy model. It is found that an optimal number of keys can be used to reduce the error. Furthermore, this thesis explores transport protocols performed on wires which feature disorder on the chemical potential and noise in the piano key tuning. The rise in the diabatic error due to these additional effects with respect to parameters-of-interest including the disorder strength, disorder correlation length, noise amplitude, and noise frequency cut-offs is characterized using numerical methods. For disorder, it is found that the error’s behaviour is principally controlled by the statistics of the minimum bulk energy gap. Spatially correlated disorder is also studied which presents non-monotonic behaviour for the error. Finally, for noise, it is found that optical transitions can cause a substantial increase in the error as the frequency modes within the noise approach the minimum energy gap.

DOI
10.82308/55754

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Bulletin of the American Physical Society•Jan 1, 2021•0 citations
Majorana zero modes (MZMs) appear as edge states in topological superconductors and are envisioned as promising candidates for application towards fault-tolerant quantum computation. Quantum information may be stored in a degenerate ground state subspace constructed from MZMs while quantum gates may be performed by braiding MZMs and exploiting their non-Abelian properties. While the topological protection of MZMs ensures that they are robust against decoherence, their dynamical manipulation can nonetheless induce undesirable transitions to excited states, manifesting as a “diabatic error.”This thesis studies the diabatic error accumulated as MZMs are transported across a superconducting wire, which is an essential building block for physical braiding schemes. The transport protocol considered relies on appropriately tuning sections (“piano keys”) of a wire between distinct topological phases via electric gates to control local chemical potentials. First, the diabatic error is examined for transport which uses multiple piano keys. The error is calculated through numerical simulation of the transport protocol and is complemented by approximate analytical expressions calculated using an effective, low-energy model. It is found that an optimal number of keys can be used to reduce the error. Furthermore, this thesis explores transport protocols performed on wires which feature disorder on the chemical potential and noise in the piano key tuning. The rise in the diabatic error due to these additional effects with respect to parameters-of-interest including the disorder strength, disorder correlation length, noise amplitude, and noise frequency cut-offs is characterized using numerical methods. For disorder, it is found that the error’s behaviour is principally controlled by the statistics of the minimum bulk energy gap. Spatially correlated disorder is also studied which presents non-monotonic behaviour for the error. Finally, for noise, it is found that optical transitions can cause a substantial increase in the error as the frequency modes within the noise approach the minimum energy gap.
Majorana Bound States (MBS) have emerged as promising candidates for robust quantum computing due to their non-Abelian statistics and topological protection. In this study, we focus on the dynamical transport of MBS in the semiconductor-superconductor (SM-SC) heterostructure via the piano key-type setup, wherein each of the keys of the wire can be tuned from topological to trivial phases. We focus on the transport of MBS under noisy conditions and evaluate the feasibility for realistic scenarios. The central emphasis of our work lies in using both numerical and analytical techniques to understand the effect of noise in inducing diabatic errors during transport and to establish scaling laws that relate these errors to the drive time. To achieve this, we derive an effective model that captures the scaling behavior in both noise-free and noisy scenarios, providing a unified framework for analyzing the transport dynamics. We investigate the optimal number of keys for both noisy and noiseless scenarios. Additionally, we explore the effects of disorder on transport dynamics, highlighting its impact on error scaling and robustness.
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.
Building a fault-tolerant quantum computer requires physical qubits with exceptionally low error rates. An exciting proposal for achieving this goal is to employ Majorana-based qubits, which may have error rates that are exponentially suppressed in macroscopic system parameters. Specifically, for Majorana qubits based on topological superconducting nanowires, errors are expected to be exponentially suppressed in the nanowire length, superconducting gap, and decreasing temperature. This thesis investigates errors in Majorana-based qubits from excited quasiparticles due to changes in chemical potentials and 1/f charge noise, and shows that such errors are not exponentially suppressed and can be quite substantial. In the first study of this thesis, we show that small temporal variations of the chemical potential can cause errors that grow linearly with the length of the nanowire. These errors stem from leakage into excited bulk quasiparticle states, which poison Majorana modes at opposite ends of the nanowire. We further demonstrate that the dynamics of this leakage are captured by the half Landau-Zener effect, which dictates its dependence on key parameters such as the superconducting gap, chemical potential variations, and dynamic changes in the spatial profile of Majorana modes. In the second study we investigate quasiparticle excitations due to high-frequency components of 1/f charge noise, which is ubiquitous in the materials surrounding the nanowire. We model 1/f noise as arising from two level fluctuators which cause chemical potential fluctuations following random telegraph noise. We demonstrate that 1/f noise causes error rates which grow linearly with the system size and are not exponentially suppressed in the superconducting gap. For existing tetron qubit designs, we estimate decoherence times to be an order of magnitude shorter than the target 1μs measurement time. This thesis also develops an algorithmic framework for applying the covariance matrix method to the time-dependent Kitaev chain model. This extends upon existing approaches, providing a key numerical tool used throughout this thesis. This work motivates engineering strategies to reduce coupling to two level systems in Majorana qubits, such as increasing the nanowire capacitance. Beyond Majoranas, a promising avenue is to investigate the impact of this quasiparticle excitation mechanism on superconducting circuits, such as superconducting qubits, resonators, and single-photon detectors.
FINDING: Majorana zero modes (MZMs) realize non-Abelian braid statistics via the Kitaev chain, with topological protection scaling exponentially in wire length, enabling topologically fault-tolerant quantum gates. | MATH: Kitaev chain Hamiltonian H=∑j[−t(cj†cj+1+h.c.)−μ(cj†cj−1/2)+Δ(cjcj+1+h.c.)] H = \sum_j [-t(c_j^\dagger c_{j+1} + h.c.) - \mu(c_j^\dagger c_j - 1/2) + \Delta(c_j c_{j+1} + h.c.)] H=∑j​[−t(cj†​cj+1​+h.c.)−μ(cj†​cj​−1/2)+Δ(cj​cj+1​+h.c.)]; Majorana operators γ2j−1=cj+cj† \gamma_{2j-1} = c_j + c_j^\dagger γ2j−1​=cj​+cj†​, γ2j=−i(cj−cj†) \gamma_{2j} = -i(c_j - c_j^\dagger) γ2j​=−i(cj​−cj†​), satisfying {γa,γb}=2δab \{\gamma_a, \gamma_b\} = 2\delta_{ab} {γa​,γb​}=2δab​; topological gap Δtop∼Δ \Delta_{\text{top}} \sim \Delta Δtop​∼Δ (superconducting gap); exponential suppression of coupling between MZMs: ϵ∝e−L/ξ \epsilon \propto e^{-L/\xi} ϵ∝e−L/ξ where ξ=ℏvF/Δ \xi = \hbar v_F / \Delta ξ=ℏvF​/Δ is the coherence length; braid group Bn B_n Bn​ generated by exchanges σi \sigma_i σi​ with σi2=−1 \sigma_i^2 = -1 σi2​=−1 (fermionic parity), yielding non-Abelian representation ρ(σi)=e±πγiγi+1/4 \rho(\sigma_i) = e^{\pm \pi \gamma_i \gamma_{i+1}/4} ρ(σi​)=e±πγi​γi+1​/4. | CONNECTION: The braid group Bn B_n Bn​ is the fundamental Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The topic of Majorana zero modes has been a focal point of the condensed matter physicscommunity in recent times due in part to their potential applications in quantum computation.Most notably, the exotic exchange statistics of Majorana zero modes form the basis oftopologically protected quantum gates. The physical exchange, or braiding, of Majorana zeromodes is often modeled in two-dimensional systems or by using networks of wires. Recentwork has established that Majorana zero modes can be braided in a purely one-dimensionalsetting with a periodic, or Floquet, drive. Specifically, the work of Bauer et al. in Ref.[1] demonstrates that a working braiding protocol can be simulated on a Kitaev chain withperiodically modulated parameters. This Floquet system gives rise to both Majorana zeromodes and Majorana pi modes, the latter of which serves as an additional degree of freedomin the braiding protocol.We reproduce the results of Ref. [1] through numerical simulation and extend theirbraiding protocol such that it includes both disorder and interactions. We demonstrate thatthe protocol remains successful when the disorder is weak compared to the parameters of thesystem. In the case of strong disorder, we demonstrate that the protocol fails. In addition,we show that the protocol is quite sensitive to the presence of interactions. We illustratethat interactions cause a severe delocalization of the Majorana zero modes leading to thefailure of the protocol