Quantum information protected by the topology of the storage medium is expected to exhibit long coherence times. Another feature is topologically protected gates generated through braiding of Majorana bound states (MBSs). However, braiding requires structures with branched topological segments which have inherent difficulties in the semiconductor–superconductor heterostructures now believed to host MBSs. In this paper, we construct quantum bits taking advantage of the topological protection and non-local properties of MBSs in a network of parallel wires, but without relying on braiding for quantum gates. The elementary unit is made from three topological wires, two wires coupled by a trivial superconductor and the third acting as an interference arm. Coulomb blockade of the combined wires spawns a fractionalized spin, non-locally addressable by quantum dots used for single-qubit readout, initialization, and manipulation. We describe how the same tools allow for measurement-based implementation of the Clifford gates, in total making the architecture universal. Proof-of-principle demonstration of topologically protected qubits using existing techniques is therefore within reach.
Surface codes represent a promising route towards universal fault-tolerant quantum computation, allowing for the protection of stored quantum states against many types of errors. This theoretical work shows that, in addition, a significantly enhanced versatility in quantum information processing is expected when qubits are encoded into the Majorana bound states in topological superconductor-semiconductor heterostructures. In view of recent experimental evidence for Majorana states in such heterostructures, a two-dimensional network of interacting Majorana states thus may feature key advantages towards the long-term goal of a functional quantum computer. The authors discuss how topologically protected logical qubits in this Majorana surface code architecture can be defined, initialized, manipulated, and read out. The physical ingredients needed to implement such operations are familiar from topologically trivial quantum devices. In particular, by employing quantum interference terms in conductance measurements, composite single-electron pumping protocols, and gate-tunable tunnel barriers, the full set of gates required for universal quantum computation can be implemented.
Universal quantum computation (UQC) using Majorana fermions on a two-dimensional topological superconducting (TS) medium remains an outstanding open problem. This is because the quantum gate set that can be generated by braiding of the Majorana fermions does not include any two-qubit gate and also no single-qubit $\ensuremath{\pi}/8$ phase gate. In principle, it is possible to create these crucial extra gates using quantum interference of Majorana fermion currents. However, it is not clear if the motion of the various order parameter defects (vortices, domain walls, etc.), to which the Majorana fermions are bound in a TS medium, can be quantum coherent. We show that these obstacles can be overcome using a semiconductor quantum wire network in the vicinity of an
s-wave superconductor, by constructing topologically protected two-qubit gates and any arbitrary single-qubit phase gate in a topologically unprotected manner, which can be error corrected using magic-state distillation. Thus our strategy, using a judicious combination of topologically protected and unprotected gate operations, realizes UQC on a quantum wire network with a remarkably high error threshold of
0.14 as compared to ${10}^{\ensuremath{-}3}$ to ${10}^{\ensuremath{-}4}$ in ordinary unprotected quantum computation.
With the discovery of Majorana quasiparticles in semiconductor-superconductor hybrid structures, topologically protected qubits have emerged as a promising contender for quantum information processing. While the construction of a universal quantum computer with topological protection likely requires significant advances in materials science, intermediate-scale devices are nearly within the reach of current technology. As a near-term milestone for topological qubits, we propose a network of topological superconductors as a simulator of a large variety of quantum spin systems, including those with frustration. Our proposal is founded on existing technology, combining advantages of semiconducting and superconducting qubits. We identify local measurement protocols that give access to information about ground and excited states as well as dynamic correlations. The topological protection of the qubits results in longer coherence times, and relaxation to the groundstate can be controlled by coupling the network an external bath. We conclude by pointing out specific applications of the quantum simulator, e.g., spin liquids, quantum criticality, and thermalization.
We describe a concrete device roadmap toward a fault-tolerant quantum computing architecture based on noise-resilient, topologically protected Majorana-based qubits. Our roadmap encompasses four generations of devices: a single-qubit device that enables a measurement-based qubit benchmarking protocol; a two-qubit device that uses measurement-based braiding to perform single-qubit Clifford operations; an eight-qubit device that can be used to show an improvement of a two-qubit operation when performed on logical qubits rather than directly on physical qubits; and a topological qubit array supporting lattice surgery demonstrations on two logical qubits. Devices that enable this path require a superconductor-semiconductor heterostructure that supports a topological phase, quantum dots and coupling between those quantum dots that can create the appropriate loops for interferometric measurements, and a microwave readout system that can perform fast, low-error single-shot measurements. We describe the key design components of these qubit devices, along with the associated protocols for demonstrations of single-qubit benchmarking, Clifford gate execution, quantum error detection, and quantum error correction, which differ greatly from those in more conventional qubits. Finally, we comment on implications and advantages of this architecture for utility-scale quantum computation.
This thesis aims to investigate the novel physics around Majorana bound states (MBSs). MBSs are quasiparticle excitations predicted to exist in topological superconductors. Due to the inherent protection they obtain from their topology, they have long been regarded as a promising setup to build a topologically protected quantum computer. Apart from this very practical application, they continue to fascinate as platforms built with MBS exhibit unique effects like nonabelian braiding and non-Fermi liquid behaviour in the topological Kondo effect.The thesis starts with a general introduction to quantum computing, highlighting both the promises and challenges of that field. Afterwards follows an introduction to systems that are candidates to host MBSs. It focuses on the theoretical proposals and experimental realizations that are most relevant to the presented work. We continue with a summary of the most fundamental effects and the setups that make the physics of MBSs accessible. The author's contributions in this field are theoretical. However, each publication comes with the explicit goal of investigating setups that make the unique physics of MBSs directly accessible in experiments. The first three papers have been peer-reviewed and were published in scientific journals. The fourth one is currently under peer review. The first two papers investigate DC transport setups connecting to a qubit based on MBSs called the MBS tetron. We show that via an effect called parity blockade, it is possible to initialize the qubit, quantify its lifetime, and determine whether it holds MBSs or trivial (normal fermionic) states. Also employing parity blockade, the second paper establishes a transport setup that enables access to MBS fusion rules, an effect directly related to nonabelian braiding. The third paper translates the concept of a MBS tetron into the newly experimentally established approach of MBSs in quantum dot-based Kitaev chains. We denote this system the poor man's Majorana tetron. We find that the poor man's Majorana tetron in a transport setup with appropriate tuning exhibits the famed topological Kondo effect. Paper~IV focuses on nonabelian braiding. It investigates the hybridization-based braiding protocol and accounts for imperfections in MBS localization, breaking the ground state degeneracy. This detrimental effect is corrected by the introduction of a correction term, followed by a full analytical solution of the transformation induced by this generalized braiding protocol.
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