Computing the topology of an arrangement of implicitly defined real algebraic plane curves~§

We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the problem at hand. Therefore, it is worth mentioning that the efficiency of the proposed method is only assured for low-degree curves.

Majorana noise model and its influence on the power spectrum

Majorana quantum computation offers a potential approach to securely manipulating and storing quantum data in a topological manner that may effectively resist the decoherence induced by local noise. However, actual Majorana qubit setups are susceptible to noise. In this study, from a quantum dynamics perspective, we develop a noise model for Majorana qubits that accounts for quasi-particle poisoning and Majorana overlapping with fluctuation. Furthermore, we focus on Majorana parity readout methodologies, specifically those leveraging an ancillary quantum dot, and carry out an indepth exploration of continuous measurement techniques founded on the quantum jump model of a quantum point contact.Utilizing these methodologies, we proceed to analyze the influence of noise on the afore-mentioned noise model, employing numerical computation to evaluate the power spectrum and frequency curve. In the culmination of our study, we put forward a strategy to benchmark the presence and detailed properties of noise in Majorana qubits.

Phase Transitions Governed by the Fifth Power of the Golden Mean and Beyond

In this contribution results from different disciplines of science were compared to show their intimate interweaving with each other having in common the golden ratio φ respectively its fifth power φ5. The research fields cover model calculations of statistical physics associated with phase transitions, the quantum probability of two particles, new physics of everything suggested by the information relativity theory (IRT) including explanations of cosmological relevance, the ε-infinity theory, superconductivity, and the Tammes problem of the largest diameter of N non-overlapping circles on the surface of a sphere with its connection to viral morphology and crystallography. Finally, Fibonacci anyons proposed for topological quantum computation (TQC) were briefly described in comparison to the recently formulated reverse Fibonacci approach using the Janičko number sequence. An architecture applicable for a quantum computer is proposed consisting of 13-step twisted microtubules similar to tubulin microtubules of living matter. Most topics point to the omnipresence of the golden mean as the numerical dominator of our world.

Realizing Majorana fermion modes in the Kitaev model

We study the possibility to realize a Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work.To achieve this we first apply a uniform[111]magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p+ip topological superconductor of spinons.We then study possible vortex binding in such system to a topologically trivial spot in the ground state.We consider two cases in the system:one is a vacancy and the other is a fully polarized spin.We show that in both cases,the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform[111]magnetic field.The distribution and asymptotic behavior of these Majorana zero modes are studied.The Majorana zero modes in both cases decay exponentially in space,and are robust against local perturbations and other Majorana zero modes far away,which makes them promising candidates for braiding in topological quantum computing.

A new representation of orientable 2-manifold polygonal surfaces for geometric modelling

Many graphics and computer-aided design applications require that the polygonal meshes used in geometric computing have the properties of not only 2-manifold but also are orientable. In this paper, by collecting previous work scattered in the topology and geometry literature, we rigorously present a theoretical basis for orientable polygonal surface representation from a modem point of view. Based on the presented basis, we propose a new combinatorial data structure that can guarantee the property of orientable 2-manifolds and is primal/dual efficient. Comparisons with other widely used data structures are also presented in terms of time and space efficiency.

Embedded Majorana islands

Mesoscopic superconducting islands hosting Majorana zero modes(MZMs),or Majorana islands in short,offer a prototype of topological qubits.In this work we investigate theoretically the model of a generic Majorana island tunneling-coupled to a singlepiece metallic substrate,hence an embedded Majorana island.We show the crucial consequences of an interplay between the topological ground states nonlocally addressed by the MZMs and the metallic bath with coherent electron propagation:on the one hand,the topological degeneracy on the Majorana island can be preserved,by virtue of the particle-hole symmetry,despite the apparent bath-induced coupling between MZMs;on the other hand,the electronic interference in the metallic bath may lead to profound alterations to the renormalization group behavior of the hybrid system towards low energy/temperature compared with conventional Kondo physics.This work serves to establish the model of embedded Majorana islands as an experimentally relevant and theoretically intriguing problem particularly in the direction of topological quantum computation.

Recent progress on non-Abelian anyons:from Majorana zero modes to topological Dirac fermionic modes

Majorana zero modes(MZMs)have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation(TQC).In addition to the Majorana scheme,some non-Majorana quasiparticles obeying non-Abelian statistics,including topological Dirac fermionic modes,have also been proposed and therefore become new candidates for TQC.In this review,we overview the non-Abelian braiding properties as well as the corresponding braiding schemes for both the MZMs and the topological Dirac fermionic modes,emphasizing the recent progress on topological Dirac fermionic modes.A topological Dirac fermionic mode can be regarded as a pair of MZMs related by unitary symmetry,which can be realized in a number of platforms,including the one-dimensional topological insulator,higher-order topological insulator,and spin superconductor.This topological Dirac fermionic mode possesses several advantages compared with its Majorana cousin,such as superconductivity-free and larger gaps.Therefore,it provides a new avenue for investigating non-Abelian physics and possible TQC.

Recent progress on Majorana in semiconductor-superconductor heterostructures—engineering and detection

Majorana zero modes(MZMs)are exotic excitations(in condensed matter systems)of fundamental scientific interest and hold great promise as the basis of fault-tolerant topological quantum computation.Though MZMs have been predicted in many platforms,their existence is still under debate.In this paper,we review the recent progress in engineering and detecting MZMs in semiconductor-superconductor heterostructures.We also briefiy review the protocols for implementing topological quantum computation by hybrid semiconductor-superconductor nanowires.