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/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=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.
We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless
px+ipy superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.
One of the main challenges for quantum computation is that while the number of gates required to perform a non-trivial quantum computation may be very large, decoherence and errors in realistic quantum architectures limit the number of physical gate operations that can be performed coherently. Therefore, an optimal mapping of the quantum algorithm into the physically available set of operations is of crucial importance. We examine this problem for a measurement-only topological quantum computer based on Majorana zero modes, where gates are performed through sequences of measurements. Such a scheme has been proposed as a practical, scalable approach to process quantum information in an array of topological qubits built using Majorana zero modes. Building on previous work that has shown that multi-qubit Clifford gates can be enacted in a topologically protected fashion in such qubit networks, we discuss methods to obtain the optimal measurement sequence for a given Clifford gate under the constraints imposed by the physical architecture, such as layout and the relative difficulty of implementing different types of measurements. Our methods also provide tools for comparative analysis of different architectures and strategies, given experimental characterizations of particular aspects of the systems under consideration. As a further non-trivial demonstration, we discuss an implementation of the surface code in Majorana-based topological qubits. We use the techniques developed here to obtain an optimized measurement sequence that implements the stabilizer measurements using only fermionic parity measurements on nearest-neighbor topological qubit islands.
We present a topologically protected teleportation protocol based on projective parity measurements between spatially separated Majorana zero modes (MZMs), eliminating the need for dynamic braiding. Unlike conventional teleportation schemes, our method preserves logical information through nonlocal encoding and suppresses decoherence exponentially with Majorana separation. We provide a rigorous mathematical framework that includes six theorems and a lemma, proving fidelity bounds, no entropy increase under ideal QND parity measurement under quantum non-demolition (QND) measurements, and compliance with the no-cloning theorem. We demonstrate that all correction operations lie within the Clifford group, enabling efficient, fault-tolerant implementation. Furthermore, we outline a scalable architecture for multi-qubit teleportation and relate our framework to recent experimental advances in quantum-dot-based Kitaev chains and superconducting nanowire platforms. These results position Majorana-based teleportation as a thermodynamically stable and experimentally viable approach to scalable quantum information transfer. All operations discussed are Clifford-only; achieving universality requires non-Clifford resources and lies outside our scope.
We analyze the topological protection of a Clifford gate set generated by braiding four Majorana zero modes (MZMs) in a two-dimensional class-D topological superconductor. Encoding a single logical qubit in four MZMs bound to
π-flux vortices, we show that elementary exchanges generate the single-qubit Clifford group up to a global phase. The residual gate error arises from finite-size Majorana hybridization and quasiparticle poisoning, scaling as
ϵ∼exp(−L/ξ), where
L is the inter-vortex separation and
ξ is the superconducting coherence length. We propose an interferometric measurement scheme to falsify this topological protection. By threading an Aharonov-Bohm loop around two MZMs, the conductance exhibits
4π-periodic oscillations with phase jumps of
π/2 per braid. Any deviation from this quantized phase beyond the
exp(−L/ξ) envelope at large
L/ξ falsifies the topological protection mechanism, indicating gap closing or quasiparticle poisoning. ## 1. Introduction Topological quantum computation seeks to process information non-locally to achieve intrinsic fault tolerance. In two-dimensional topological superconductors belonging to symmetry class D (such as a
px+ipy paired state or a proximitized semiconductor heterostructure), Majorana zero modes (MZMs) bind to
π-flux vortices. Exchanging these vortices implements non-Abelian unitaries on the degenerate ground-state manifold. The central question addressed here is whether braiding four MZMs generates a topologically protected Clifford gate set with a residual error rate scaling as
exp(−L/ξ), and whether this claim is falsifiable via interferometric measurement of the braiding phase. ## 2. Background The non-Abelian statistics of MZMs in class-D superconductors were established by Ivanov (2001), demonstrating that exchanging vortices realizes a braid group representation with
π/2 rotations in Majorana Hilbert space [arXiv:cond-mat/0105040]. The topologi
We propose and analyze a physical system capable of performing topological quantum computation with Majorana zero modes (MZMs) in a one-dimensional topological superconductor (1DTS). One of the leading methods to realize quantum gates in a 1DTS is to use T junctions, which allows one to maneuver MZMs in such a manner as to achieve braiding. In this paper, we propose a scheme for implementing quantum logical gates in a purely one-dimensional geometry without T junctions, instead replacing it with an auxiliary qubit. This has the additional benefit of introducing a non-Clifford gate, corresponding to one- and two-logical-qubit
Z rotations. We first design a topologically protected logical
Z gate based entirely on local interactions within the 1DTS. Using an auxiliary qubit coupled to the topological superconductors, we extend the
Z gate to non-Clifford single- and multiqubit arbitrary rotations with partial topological protection. Finally, to perform universal quantum computing, we introduce a scheme for performing arbitrary unitary rotations, albeit without topological protection. We develop a formalism based on unitary braids which creates transitions between different topological phases of the 1DTS system. The unitary formalism can be simply converted to an equivalent adiabatic scheme, which we numerically simulate, and we show that high-fidelity operations should be possible with reasonable parameters.