scholarly journals Polygon sign rules of Majorana fermions in two-dimensional topological superconductors

2016 ◽  
Vol 30 (27) ◽  
pp. 1650213
Author(s):  
Qiu-Bo Cheng ◽  
Jing He ◽  
Jing Yu ◽  
Xiao-Ming Zhao ◽  
Su-Peng Kou
Keyword(s):  
Quantum Computation ◽  
Numerical Approach ◽  
Majorana Fermions ◽  
Two Dimensional ◽  
Line Defect ◽  
Topological Superconductor ◽  
Topological Superconductors ◽  
Topological Quantum ◽  
Different Types ◽  
Topological Quantum Computation

Recently, Majorana fermions (MFs) have attracted intensive attention due to their exotic statistics and possible applications in topological quantum computation. They are proposed to exist in various two-dimensional (2D) topological systems, such as [Formula: see text] topological superconductor (SC) and nanowire–superconducting hybridization system. In this paper, we point out that Majorana fermions in different topological systems obey different types of polygon sign rules. A numerical approach is described to identify the type of polygon sign rule of the Majorana fermions. Applying the approach to study two 2D topological systems, we find that vortex-induced Majorana fermions obey topological polygon sign rule and line-defect-induced Majorana fermions obey normal polygon sign rule.

scholarly journals Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm

2021 ◽  
Author(s):  
Niccolò Traverso Ziani ◽  
Lucia Vigliotti ◽  
Matteo Carrega ◽  
Fabio Cavaliere
Keyword(s):  
Quantum Computation ◽  
Bound States ◽  
Research Activity ◽  
One Dimensional ◽  
Majorana Bound States ◽  
Topological Superconductors ◽  
Topological Quantum ◽  
Tamm State ◽  
Topological Quantum Computation ◽  
Intense Research

Majorana bound states in topological superconductors have attracted intense research activity in view of applications in topological quantum computation. However, they are not the only example of topological bound states that can occur in such systems. We here study a model in which both Majorana and Tamm bound states compete. We show both numerically and analytically that, surprisingly, the Tamm state remains partially localized even when the spectrum becomes gapless. Despite this fact, we demonstrate that the Majorana polarization shows a clear transition between the two regimes.

scholarly journals Witness algebra and anyon braiding

2020 ◽  
Vol 30 (3) ◽  
pp. 234-270
Author(s):  
Andreas Blass ◽  
Yuri Gurevich
Keyword(s):  
Quantum Computation ◽  
Category Theory ◽  
Two Dimensional ◽  
Mathematical Basis ◽  
Tensor Categories ◽  
Topological Quantum ◽  
Modular Tensor Categories ◽  
Topological Quantum Computation

AbstractTopological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.

scholarly journals Topological quantum computation based on chiral Majorana fermions

2018 ◽  
Vol 115 (43) ◽  
pp. 10938-10942 ◽  
Author(s):  
Biao Lian ◽  
Xiao-Qi Sun ◽  
Abolhassan Vaezi ◽  
Xiao-Liang Qi ◽  
Shou-Cheng Zhang
Keyword(s):  
Quantum Computation ◽  
Edge State ◽  
Majorana Fermion ◽  
Majorana Fermions ◽  
Hybrid Device ◽  
Topological Quantum ◽  
Ring Junction ◽  
Phase Gate ◽  
Topological Quantum Computation ◽  
Junction Conductance

The chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain 2D topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of a quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, the Majorana zero mode in the bulk of the corresponding topological matter, is known to be applicable in topological quantum computations. Here we show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes and propose a platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can use quantum coherent chiral Majorana fermions to implement the Hadamard gate and the phase gate, and the junction conductance yields a natural readout for the qubit state.

scholarly journals Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1100
Author(s):  
Lucia Vigliotti ◽  
Fabio Cavaliere ◽  
Matteo Carrega ◽  
Niccolò Traverso Ziani
Keyword(s):  
Quantum Computation ◽  
Bound States ◽  
Research Activity ◽  
One Dimensional ◽  
Majorana Bound States ◽  
Topological Superconductors ◽  
Topological Quantum ◽  
Tamm State ◽  
Topological Quantum Computation ◽  
Intense Research

Majorana bound states in topological superconductors have attracted intense research activity in view of applications in topological quantum computation. However, they are not the only example of topological bound states that can occur in such systems. Here, we study a model in which both Majorana and Tamm bound states compete. We show both numerically and analytically that, surprisingly, the Tamm state remains partially localized even when the spectrum becomes gapless. Despite this fact, we demonstrate that the Majorana polarization shows a clear transition between the two regimes.

scholarly journals Massless Majorna bispinors and two-qubit entangled states

2020 ◽  
Vol 17 (2 Jul-Dec) ◽  
pp. 115
Author(s):  
R. Romero
Keyword(s):  
Quantum Computation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Computation Model ◽  
Linearly Independent ◽  
Topological Quantum ◽  
Different Types ◽  
Logical Qubits ◽  
The Common ◽  
Topological Quantum Computation

This is a pedagogical paper, where bispinors solutions to the four-dimensional massless Dirac equation are considered in relativistic quantum mechanics and in quantum computation, taking advantage of the common mathematical description of four-dimensional spaces. First, Weyl and massless Majorana bispinors are shown to be unitary equivalent, closing a gap in the literature regarding their equivalence. A discrepancy in the number of linearly independent solutions reported in the literature is also addressed. Then, it is shown that Weyl bispinors are algebraically equivalent to two-qubit direct product states, and that the massless Majorana bispinors are algebraically equivalent to maximally entangled sates (Bell states), with the transformations relating the two bispinors types acting as entangling gates in quantum computation. Different types of entangling gates are presented, highlighting a set that fulfills the required properties for Majorana zero mode operators in topological quantum computation. Based on this set, a general topological quantum computation model with four Majorana operators is presented, which exhibits all the required technical and physical properties to obtain entanglement of two logical qubits from tpological operations.

scholarly journals Optimizing Clifford gate generation for measurement-only topological quantum computation with Majorana zero modes

2020 ◽  
Vol 8 (6) ◽  
Author(s):  
Alan Tran ◽  
Alex Bocharov ◽  
Bela Bauer ◽  
Parsa Bonderson
Keyword(s):  
Quantum Computation ◽  
Nearest Neighbor ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Zero Modes ◽  
Topological Quantum ◽  
Different Types ◽  
Computer Based ◽  
Surface Code ◽  
Topological Quantum Computation

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.

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