Topological qubits based on non-Abelian Majorana bound states (MBSs) are protected by parity which is challenged by quasiparticle poisoning (QPP). In this work, we show how QPP affects transport through a pair of coupled MBSs weakly connected to two current leads, using an open system approach and full counting statistics. We find that the correct low-energy physics requires one to include next to leading order tunneling events in the coupling to the leads. In particular, our results show that QPP causes super-Poissonian local shot noise with a Fano factor that diverges with decreasing bias voltage, while the current and the nonlocal (cross) noise are only little affected. We explain that these features are a direct consequence of the nature of MBSs being their own antiparticle making noise measurements a viable tool to search for MBSs in the presence of QPP.
Topological qubits based on non-Abelian Majorana bound states (MBSs) are protected by parity which is challenged by quasiparticle poisoning (QPP). In this work, we show how QPP affects transport through a pair of coupled MBSs weakly connected to two current leads, using an open system approach and full counting statistics. We find that the correct low-energy physics requires one to include next to leading order tunneling events in the coupling to the leads. In particular, our results show that QPP causes super-Poissonian local shot noise with a Fano factor that diverges with decreasing bias voltage, while the current and the nonlocal (cross) noise are only little affected. We explain that these features are a direct consequence of the nature of MBSs being their own antiparticle making noise measurements a viable tool to search for MBSs in the presence of QPP.
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.
Majorana-based quantum computing seeks to use the non-local nature of Majorana zero modes to store and manipulate quantum information in a topologically protected way. While noise is anticipated to be significantly suppressed in such systems, finite temperature and system size result in residual errors. In this work, we connect the underlying physical error processes in Majorana-based systems to the noise models used in a fault tolerance analysis. Standard qubit-based noise models built from Pauli operators do not capture leading order noise processes arising from quasiparticle poisoning events, thus it is not obvious a priori that such noise models can be usefully applied to a Majorana-based system. We develop stochastic Majorana noise models that are generalizations of the standard qubit-based models and connect the error probabilities defining these models to parameters of the physical system. Using these models, we compute pseudo-thresholds for the d=5 Bacon-Shor subsystem code. Our results emphasize the importance of correlated errors induced in multi-qubit measurements. Moreover, we find that for sufficiently fast quasiparticle relaxation the errors are well described by Pauli operators. This work bridges the divide between physical errors in Majorana-based quantum computing architectures and the significance of these errors in a quantum error correcting code.
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.
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.
The possibility of engineering artificial Kitaev chains in arrays of quantum dots coupled via narrow superconducting regions has emerged as an attractive way to overcome the disorder issues that complicate the realization and detection of topological superconducting phases in other platforms. Although a true topological phase would require long chains, a two-site chain realized in a double quantum dot can already be tuned to points in parameter space where it hosts zero-energy states that seem identical to the Majorana bound states that characterize a topological phase. These states have been named “poor man’s Majorana bound states” (PMMs) because they lack formal topological protection. In this work, we propose a pathway for next-generation experiments on PMMs. The pathway starts with experiments to characterize a single pair of PMMs by measuring the Majorana quality and then moves on to initialization and readout of the parity of a PMM pair, which allows the measurement of quasiparticle poisoning times. The next step is to couple two PMM systems to form a qubit. We discuss measurements of the coherence time of such a qubit, as well as a test of Majorana fusion rules in the same setup. Finally, we propose and analyze three different types of braidinglike experiments that require more complex device geometries. Our conclusions are supported by calculations based on a realistic model with interacting and spinful quantum dots, as well as by simpler models to gain physical insight. Our calculations show that it is indeed possible to demonstrate non-Abelian physics in minimal two-site Kitaev chains despite the lack of a true topological phase. However, our findings also reveal that doing so requires some extra care, appropriately modified protocols, and awareness of the details of this particular platform. Published by the American Physical Society 2024
We study the dissipative dynamics of a Majorana tetron qubit in the presence of extrinsic quasiparticle poisoning due to the coupling to external leads. From the Bloch-Redfield equation describing a finite-size topological superconductor hosting four Majorana zero modes and tunnel-coupled to fermionic reservoirs, we recover analytical expressions for the decoherence rate of a Majorana qubit at arbitrary values of the charging energy. The analysis shows that the exponential suppression of the decoherence rate is gradually removed by the energy splitting of the qubit states. These results can be useful to understand time-domain experimental data in Majorana qubit prototypes.