The quest to identify and observe Majorana fermions in physics and condensed-matter systems remains an important challenge. Here, we introduce a qubit (spin- 1 / 2 ) from the occurrence of two delocalized zero-energy Majorana fermions in a model of two spins- 1 / 2 on the Bloch sphere within the fractional one-half topological state. We address specific protocols in time with circularly polarized light and the protection of this delocalized spin- 1 / 2 state related to quantum information protocols. We also show how disorder can play a positive and important role allowing singlet-triplet transitions and resulting in an additional elongated region for the fractional phase, demonstrating the potential of this platform related to applications in topologically protected quantum information. We generalize our approach with an array and Majorana fermions at the edges in a ring geometry. Published by the American Physical Society 2024
The quest to identify and observe Majorana fermions in physics and condensed-matter systems remains an important challenge. Here, we introduce a qubit (spin- 1 / 2 ) from the occurrence of two delocalized zero-energy Majorana fermions in a model of two spins- 1 / 2 on the Bloch sphere within the fractional one-half topological state. We address specific protocols in time with circularly polarized light and the protection of this delocalized spin- 1 / 2 state related to quantum information protocols. We also show how disorder can play a positive and important role allowing singlet-triplet transitions and resulting in an additional elongated region for the fractional phase, demonstrating the potential of this platform related to applications in topologically protected quantum information. We generalize our approach with an array and Majorana fermions at the edges in a ring geometry. Published by the American Physical Society 2024
Majorana fermions arise in a broad range of systems in condensed matter physics, not only as a useful abstraction but also as emergent quasiparticles, e.g., at the boundaries of topological insulators. In this dissertation, we consider two distinct settings wherein Majorana fermions play a central role: (i) non-abelian Kitaev spin liquids and (ii) entanglement dynamics in monitored quantum circuitsThe Kitaev spin liquid is a paradigmatic model featuring non-abelian anyons and a chiral Majorana edge mode when time-reversal symmetry is broken. To this end, it presents an alluring platform for realizing topologically protected quantum information. However, experimental efforts to identify this putative phase have been hampered both by the difficulty of probing the charge neutral quasiparticles and by the presence of phonons and other deviations from the idealized toy model. Here we examine how judicious device design may suppress bulk phonon contributions to thermal transport, allowing for refined observation of transport mediated by the chiral Majorana edge mode. Moreover, we show that the bulk topological order can be revealed through universal features in the tunneling conductance across a point contact. Looking forward, we develop a roadmap for near-term experimental devices to demonstrate robust control of topologically protected qubits encoded in the bulk anyons.Much as Majorana models form the building blocks for symmetry-protected topological order, Majorana circuits provide a powerful framework for studying the entanglement dynamics and phases of monitored quantum circuits. Here we show that the entanglement dynamics of Majorana circuits can be mapped exactly to the statistical mechanics of classical loop models. Harnessing this framework, we undertake a precise numerical characterization of the universality of entanglement transitions in symmetry classes BDI and D in both (1+1) and (2+1)-dimensional circuits. Furthermore, we employ duality relations for loop models to construct an analytically solvable interacting circuit which features not only the usual percolation criticality but also Berezinskii-Kosterlitz-Thouless critical lines and a tricritical point with Ising universality. Lastly, by generalizing to colored loop models for Majorana circuits with classical control, we provide a blueprint for engineering even more diverse entanglement dynamics which may include Hilbert-space fragmentation and subdiffusive transport.
D eep in the heart of the topological world lives the Majorana fermion. There are several reasons the Majorana quasiparticle occu- pies a focal point within topological quantum matter. Of course, conceptually, it is a classical example of boundary mode that reveals the topological properties of the bulk, in this case a topological superconductor; an example that captures extremely well the very character of topological matter: simple yet subtle, hiding in plain view. But what enables the Majorana to play an important symbolic role is its conspicuous position at the crossroad that defines the new paradigm underlying the study of topological quantum matter. A child of particle physics, the Majorana was reborn in condensed matter physics and grew under the guidance of quantum computation. Lost before it was even found, it is fervently searched for in solid state and cold atom systems because it carries the promise to revolutionize the manner in which we process information and understand the foundations of the quantum world. Whether or not the Majorana will ever fulfill this promise is a minor detail; its main role is that of a legend, to give hope and move things forward. Practically, the search for zero-energy Majorana modes will play an important role in achieving the key condition for getting experimental access to topological quantum matter and quantum computation: an unprecedented level of control over quantum systems. At the time of this writing (spring 2016), the technological impact of these developments cannot be clearly foreseen.
Topological qubits composed of unpaired Majorana zero modes are under intense experimental and theoretical scrutiny in efforts to realize practical quantum computation schemes. In this work, we show that the minimum four Majorana zero modes required for a topological qubit according to braiding schemes and control of entanglement for gate operations are inherent to multiplicative topological phases, which realize symmetry-protected tensor products—and maximally entangled Bell states—of unpaired Majorana zero modes. We construct and characterize both one-dimensional and two-dimensional multiplicative topological phases with two parent Kitaev chain Hamiltonians. We furthermore characterize topology in the bulk and on the boundary with established methods while also introducing techniques to overcome challenges in characterizing multiplicative topology. In the process, we explore the potential of these multiplicative topological phases for an alternative to braiding-based topological quantum computation schemes, in which gate operations are performed through topological phase transitions. Published by the American Physical Society 2024
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.
Spin-triplet p-wave superfluids of single-species (or spin-polarized) fermionic atoms is a topological superfluid. In 2D such a superfluid supports zero-energy topological Majorana fermion excitations in order parameter defects such as vortices and sample edges. In 3D these superfluids support topologically protected Dirac points in the bulk spectrum and flat surface Majorana arcs. Despite the promise, creating a spin-triplet p-wave single-species fermionic superfluid either from direct p-wave Feshbach resonance or indirectly from the recently proposed artificial spin-orbit and Zeeman couplings can be technologically challenging. Here we show that such topological superfluids can be far more simply created by using the Kohn-Luttinger effect applied to two species of spin-polarized fermions with a density imbalance. We discuss how the topological Majorana excitations of the resulting superfluids can be identified using recently-developed experimental techniques.
A Majorana fermion is a fermionic particle that is its own antiparticle. Since the theoretical discovery in 1937, the exotic particle has long been searched in particle physics. In the last few decades, however, it has attracted renewed interest in condensed matter physics, where it can be realized as an elementary excitation (quasiparticle) in quantum states of matter. In this review, we discuss another platform for Majorana fermions, the quantum spin liquid, in which interacting magnetic moments remain disordered down to the lowest temperature under strong quantum fluctuations. They are characterized by topological entanglement and fractional excitations, whose possible application to topological quantum computation is recently discussed intensively. As a prime candidate for such exotic states, we here focus on the Kitaev magnets, a subgroup of the spin-orbit Mott insulators. After a brief overview of the Kitaev model and the fractionalization of spins in the exact ground state, we review recent explosive development in this rapidly growing field, with a focus on numerical solutions of the Kitaev model at finite temperatures and the comparison with experiments. The key concept is thermal fractionalization --- two types of fractional excitations manifest themselves at largely different temperatures. This leads to distinct thermodynamics and spin dynamics in a variety of experimentally measurable quantities. We discuss such peculiar behaviors as the signatures of fractional quasiparticles, in careful comparison with the available experimental data for the candidate materials of the Kitaev magnets. Our review gives an overview of the current status of the identification of Majorana fermions in the Kitaev magnets, which would serve as a basis for further experimental and theoretical studies toward the manipulation of the exotic particles for topological quantum computation.