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arXiv:1512.03486 · Physical review. B./Physical review. B

Quantum information sharing between topologically distinct platforms

Chang-Yu Hou, Gil Refael, Kirill Shtengel

Published Dec 5, 2016• 0 citations
arXivHTMLPDF

Abstract

Can topological quantum entanglement between anyons in one topological medium ``stray'' into a different, topologically distinct medium? In other words, can quantum information encoded nonlocally in the combined state of non-Abelian anyons be shared between two distinct topological media? For one-dimensional topological superconductors with Majorana bound states at the end of system, the quantum information store in those Majorana bound states can be transfered by directly coupling nearby Majorana bound states. However, coupling of two one-dimensional Majorana states will produce a gap, indicating that distinct topological regions of one-dimensional wires unite into a single topological region through the information transfer process. In this paper, we consider a setup with two two-dimensional $p$-wave superconductors of opposite chirality adjacent to each other. Even two comoving chiral modes at the domain wall between them cannot be gapped through interactions; we demonstrate that information encoded in the fermionic parity of two Majorana zero modes, originally within the same superconducting domain, can be shared between the domains or moved entirely from one domain to another provided that vortices can tunnel between them in a controlled fashion.

DOI
10.1103/physrevb.94.235113

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FINDING: Topological quantum computing leverages anyonic quasiparticles and braid group representations to encode information non-locally, with Microsoft's Majorana 1 chip claiming topological qubit hardware based on Majorana zero modes. | MATH: Braid group \( B_n \) generators \( \sigma_i \) satisfy \( \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \) and \( \sigma_i \sigma_j = \sigma_j \sigma_i \) for \( |i-j|>1 \). Fibonacci anyons: fusion rule \( \tau \otimes \tau = 1 \oplus \tau \), quantum dimension \( d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 \). Non-Abelian statistics: unitary braid matrices \( U(\sigma_i) \) with entries involving \( e^{i\pi/5} \), \( \phi^{-1} \), \( \phi^{-1/2} \). Majorana zero modes: \( \gamma_i^\dagger = \gamma_i \), \( \{\gamma_i, \gamma_j\} = 2\delta_{ij} \), parity \( P = i^k \prod \gamma_i \). Topological protection: energy gap \( \Delta \) vs. error rate \( \sim e^{-\Delta/k_B T} \). | CONNECTION: Fibonacci anyons directly enc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Can topological quantum entanglement between anyons in one topological medium ``stray'' into a different, topologically distinct medium? In other words, can quantum information encoded nonlocally in the combined state of non-Abelian anyons be shared between two distinct topological media? For one-dimensional topological superconductors with Majorana bound states at the end of system, the quantum information store in those Majorana bound states can be transfered by directly coupling nearby Majorana bound states. However, coupling of two one-dimensional Majorana states will produce a gap, indicating that distinct topological regions of one-dimensional wires unite into a single topological region through the information transfer process. In this paper, we consider a setup with two two-dimensional ppp-wave superconductors of opposite chirality adjacent to each other. Even two comoving chiral modes at the domain wall between them cannot be gapped through interactions; we demonstrate that information encoded in the fermionic parity of two Majorana zero modes, originally within the same superconducting domain, can be shared between the domains or moved entirely from one domain to another provided that vortices can tunnel between them in a controlled fashion.
One-dimensional topological superconductors are known to host Majorana zero modes at domain walls terminating the topological phase. Their non-Abelian nature allows for processing quantum information by braiding operations that are insensitive to local perturbations, making Majorana zero modes a promising platform for topological quantum computation. Motivated by the ultimate goal of executing quantum-information processing on a finite time scale, we study domain walls moving at a constant velocity. We exploit an effective Lorentz invariance of the Hamiltonian to obtain an exact solution of the associated quasiparticle spectrum and wave functions for arbitrary velocities. Essential features of the solution have a natural interpretation in terms of the familiar relativistic effects of Lorentz contraction and time dilation. We find that the Majorana zero modes remain stable as long as the domain wall moves at subluminal velocities with respect to the effective speed of light of the system. However, the Majorana bound state dissolves into a continuous quasiparticle spectrum after the domain wall propagates at luminal or even superluminal velocities. This relativistic catastrophe implies that there is an upper limit for possible braiding frequencies even in a perfectly clean system with an arbitrarily large topological gap. We also exploit our exact solution to consider domain walls moving past static impurities present in the system.
Proximity-induced superconductivity in low-dimensional systems offers a powerful pathway to engineer topological superconducting phases in, otherwise, non-superconducting systems. These exotic phases are of fundamental and technological interest due to the presence of robust zero-energy modes, the Majorana bound states. In this work, we propose a theoretical framework to realize Majorana bound states from the edge states of a two-dimensional semi-Dirac system. This anisotropic system, under specific conditions, can host non-chiral edge states that propagate only along particular edges, effectively forming separated one-dimensional channels. We show that the interplay between Rashba spin-orbit coupling and a Zeeman field on this setup provides the right conditions to get an effective p-wave pairing between the edge states by proximity with a s-wave superconductor. In finite geometries, each edge can independently undergo a topological phase transition into a one-dimensional topological superconductor and give rise to four zero-energy modes localized at the strip corners. At low energies, the edge states subspace admits a description in terms of coupled Kitaev chains, providing a clear picture of the origin, robustness, and tunability of the corner Majorana modes. Our results establish semi-Dirac materials as a natural platform for realizing Majorana modes in two dimensions without relying on engineered nanostructures, vortices, or crystalline higher-order topology.
We propose a scheme to realize coherent quantum information transfer between topological and conventional charge qubits. We first consider a hybrid system where a quantum dot(QD) is tunnel-coupled to a semiconductor Majorana-hosted nanowire(MNW) via using gated control as a switch, the information encoded in the superposition state of electron empty and occupied state can be transferred to each other through choosing the proper interaction time to make measurements.Then we consider another system including a double QDs and a pair of parallel MNWs, it is shown that the entanglement information transfer can be realized between the two kinds of systems. We also realize long distance quantum information transfer between two quantum dots separated by an MNW, by making use of the nonlocal fermionic level formed with the pared Majorana feimions(MFs) emerging at the two ends of the MNW. Furthermore, we analyze the teleportationlike electron transfer phenomenon predicted by Tewari et al. [Phys. Rev. Lett. 100, 027001(2008)] in our considered system.Interestingly, we find that this phenomenon exactly corresponds to the case that the information encoded in one QD just returns back to its original place during the dynamical evolution of the combined system from the perspective of quantum state transfer.
We propose a scheme to realize coherent quantum information transfer between topological and conventional charge qubits. We first consider a hybrid system where a quantum dot (QD) is tunnel-coupled to a semiconductor Majorana-hosted nanowire (MNW) via using gated control as a switch, the information encoded in the superposition state of electron empty and occupied state can be transferred to each other through choosing the proper interaction time to make measurements. Then we consider another system including a double QDs and a pair of parallel MNWs, it is shown that the entanglement information transfer can be realized between the two kinds of systems. We also realize long distance quantum information transfer between two quantum dots separated by an MNW, by making use of the nonlocal fermionic level formed with the pared Majorana feimions (MFs) emerging at the two ends of the MNW. Furthermore, we analyze the teleportationlike electron transfer phenomenon predicted by Tewari et al. [ Phys. Rev. Lett. 100 , 027001 (2008)] in our considered system. Interestingly, we find that this phenomenon exactly corresponds to the case that the information encoded in one QD just returns back to its original place during the dynamical evolution of the combined system from the perspective of quantum state transfer.
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 sss-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 0.140.140.14 as compared to ${10}^{\ensuremath{-}3}$ to ${10}^{\ensuremath{-}4}$ in ordinary unprotected quantum computation.
FINDING: Topological quantum computing leverages anyonic quasiparticles and braid group representations to encode information non-locally, with Microsoft's Majorana 1 chip claiming topological qubit hardware based on Majorana zero modes. | MATH: Braid group Bn B_n Bn​ generators σi \sigma_i σi​ satisfy σiσi+1σi=σi+1σiσi+1 \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} σi​σi+1​σi​=σi+1​σi​σi+1​ and σiσj=σjσi \sigma_i \sigma_j = \sigma_j \sigma_i σi​σj​=σj​σi​ for ∣i−j∣>1 |i-j|>1 ∣i−j∣>1. Fibonacci anyons: fusion rule τ⊗τ=1⊕τ \tau \otimes \tau = 1 \oplus \tau τ⊗τ=1⊕τ, quantum dimension dτ=ϕ=(1+5)/2≈1.618 d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 dτ​=ϕ=(1+5​)/2≈1.618. Non-Abelian statistics: unitary braid matrices U(σi) U(\sigma_i) U(σi​) with entries involving eiπ/5 e^{i\pi/5} eiπ/5, ϕ−1 \phi^{-1} ϕ−1, ϕ−1/2 \phi^{-1/2} ϕ−1/2. Majorana zero modes: γi†=γi \gamma_i^\dagger = \gamma_i γi†​=γi​, {γi,γj}=2δij \{\gamma_i, \gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​, parity P=ik∏γi P = i^k \prod \gamma_i P=ik∏γi​. Topological protection: energy gap Δ \Delta Δ vs. error rate ∼e−Δ/kBT \sim e^{-\Delta/k_B T} ∼e−Δ/kB​T. | CONNECTION: Fibonacci anyons directly enc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com