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arXiv:2009.07590 · Physical Review Letters

Emulating Quantum Teleportation of a Majorana Zero Mode Qubit

He-Liang Huang, Marek Narożniak, Futian Liang, Youwei Zhao, Juno Clark, Ming Gong, Yulin Wu, Shiyu Wang, Jin Lin, Yu Xu, Hui Deng, Hao Rong, Jonathan P. Dowling, Cheng-Zhi Peng, Tim Byrnes, Xiaobo Zhu, Jian-Wei Pan

Published Mar 3, 2021• 48 citations
arXivHTMLPDF

Abstract

Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we perform a quantum simulation of topological quantum computing, by teleporting a qubit encoded in the Majorana zero modes of a Kitaev chain. The quantum simulation is performed by mapping the Kitaev chain to its equivalent spin version and realizing the ground states in a superconducting quantum processor. The teleportation transfers the quantum state encoded in the spin-mapped version of the Majorana zero mode states between two Kitaev chains. The teleportation circuit is realized using only braiding operations and can be achieved despite being restricted to Clifford gates for the Ising anyons. The Majorana encoding is a quantum error detecting code for phase-flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from 70.76±0.35% to 84.60±0.11%, well beyond the classical bound in either case.

DOI
10.1103/physrevlett.126.090502

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Quantum error correction is widely considered to be an essential ingredient for overcoming decoherence and achieving large-scale quantum computation. Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we demonstrate in a superconducting quantum processor that the spin-mapped version of the Majorana zero modes can be used to perform quantum teleportation. The teleportation transfers the quantum state encoded on two-qubit Majorana zero mode states between two Kitaev chains, using only braiding operations. The Majorana encoding is a quantum-error-detecting code for phase flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from $70.76 \pm 0.35 \% $ to $84.60 \pm 0.11 \%$, well beyond the classical bound in either case.

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FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group \( B_n \) generators satisfy \( \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \); Majorana operators \( \gamma_i \) satisfy \( \{\gamma_i, \gamma_j\} = 2\delta_{ij} \); fusion rules for Ising anyons: \( \sigma \times \sigma = 1 + \psi \), \( \sigma \times \psi = \sigma \), \( \psi \times \psi = 1 \). | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at \( q = e^{i\pi/5} \), where the Jones polynomial yields \( \tau = (1+\sqrt{5})/2 \approx 1.618 \); Fibonacci anyons have fusion rules \( \tau \times \tau = 1 + \tau \), directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Jul 22, 2026•0 citations

Non-Abelian braiding of Majorana-like edge states and quantum gates in electric circuits

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Majorana fermions subject to the non-Abelian braid group are believed to be the basic ingredients of future topological quantum computations. In this work, we propose to simulate Majorana fermions of the Kitaev model in electric circuits. We generate an arbitral number of segments in a Kitaev chain which are in the topological phase. A pair of Majorana-like edge states emerge at the edges of a topological sector, which is observed by impedance peaks. When the Kitaev chain contains $N$ topological sectors, the degeneracy of the zero-admittance states is $2^{N}$. A set of AC voltages along the chain plays a role of the wave function. An \textit{LC} resonator attached to each Majorana-like edge state registers the phase and the absolute value of the voltage. One qubit consists of the in-phase and opposite-phase states made of two \textit{LC} resonators installed in one topological sector. It is possible to braid any pair of neighboring edge states with the aid of T-junction geometry. By calculating the Berry phase acquired by their eigenfunctions, the braiding is shown to generate one-qubit and two-qubit unitary operations. We explicitly construct Clifford quantum gates based on them. The present scheme may simulate topological quantum computations based on electric circuits.

•Jul 16, 2019•0 citations
Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we perform a quantum simulation of topological quantum computing, by teleporting a qubit encoded in the Majorana zero modes of a Kitaev chain. The quantum simulation is performed by mapping the Kitaev chain to its equivalent spin version and realizing the ground states in a superconducting quantum processor. The teleportation transfers the quantum state encoded in the spin-mapped version of the Majorana zero mode states between two Kitaev chains. The teleportation circuit is realized using only braiding operations and can be achieved despite being restricted to Clifford gates for the Ising anyons. The Majorana encoding is a quantum error detecting code for phase-flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from 70.76±0.35% to 84.60±0.11%, well beyond the classical bound in either case.
Quantum error correction is widely considered to be an essential ingredient for overcoming decoherence and achieving large-scale quantum computation. Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we demonstrate in a superconducting quantum processor that the spin-mapped version of the Majorana zero modes can be used to perform quantum teleportation. The teleportation transfers the quantum state encoded on two-qubit Majorana zero mode states between two Kitaev chains, using only braiding operations. The Majorana encoding is a quantum-error-detecting code for phase flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from $70.76 \pm 0.35 \% $ to 84.60±0.11%84.60 \pm 0.11 \%84.60±0.11%, well beyond the classical bound in either case.
Topological superconductivity could host quasiparticles that are key candidates for fault-tolerant quantum computation due to their immunity to noise as they obey non-Abelian exchange statistics. For example, in the case of Majorana Zero Modes (MZM), braiding enables two topologically protected quantum gates. While their direct manipulation in solid-state systems remains experimentally challenging, digital emulation of MZM behavior has provided insight as well as a deeper understanding of controlling these topological quantum systems. This emulation is typically accomplished by mapping the topological and trivial phases of a Majorana system to ferromagnetic and paramagnetic Hamiltonians of a spin-glass model. This approach usually relies on adiabatic evolution of superconducting Hamiltonians, which require circuits with very large depths. In this work, we present a resource-efficient method to emulate MZM braiding in a trijunction geometry using a quantum processor. We introduce direct braiding operators which simulate the evolution more efficiently, reducing the quantum gate overhead. We then further generalize this method to emulate braiding operations in extended trijunction architectures based on Kitaev chains.
Topological superconductivity could host quasi-particles that are key candidates for fault-tolerant quantum computation due to their immunity to noise as they obey non-Abelian exchange statistics. For example, in the case of Majorana zero modes (MZM), braiding enables two topologically protected quantum gates. While their direct manipulation in solid-state systems remains experimentally challenging, digital emulation of MZM behavior has provided insight as well as a deeper understanding of controlling these topological quantum systems. This emulation is typically accomplished by mapping the topological and trivial phases of a Majorana system to ferromagnetic and paramagnetic Hamiltonians of a spin-glass model. This approach usually relies on adiabatic evolution of superconducting Hamiltonians, which require circuits with very large depths. In this work, we present a resource-efficient method to emulate MZM braiding in a trijunction geometry using a quantum processor. We introduce direct braiding operators which simulate the evolution more efficiently, reducing the quantum gate overhead. We then further generalize this method to emulate braiding operations in extended trijunction architectures based on Kitaev chains.
Topological quantum computation aims to employ anyonic quasiparticles with exotic braiding statistics to encode and manipulate quantum information in a fault-tolerant way. Majorana zero modes are experimentally the simplest realisation of anyons that can non-trivially process quantum information. However, their braiding evolutions, necessary for realising topological gates, still remain beyond current technologies. Here we report the experimental encoding of four Majorana zero modes in an all-optical quantum simulator that give rise to a fault-tolerant qubit. We experimentally simulate their braiding and demonstrate both the non-Abelian character and the topological nature of the resulting geometric phase. We realise a full set of topological and non-topological gates that can arbitrarily rotate the encoded qubit. As an application, we implement the Deutsch-Jozsa algorithm exclusively by topological gates. Our experiment indicates the intriguing possibility of the experimental simulation of Majorana-based quantum computation with scalable technologies.
FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group Bn B_n Bn​ generators 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​; Majorana operators γi \gamma_i γi​ satisfy {γi,γj}=2δij \{\gamma_i, \gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​; fusion rules for Ising anyons: σ×σ=1+ψ \sigma \times \sigma = 1 + \psi σ×σ=1+ψ, σ×ψ=σ \sigma \times \psi = \sigma σ×ψ=σ, ψ×ψ=1 \psi \times \psi = 1 ψ×ψ=1. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at q=eiπ/5 q = e^{i\pi/5} q=eiπ/5, where the Jones polynomial yields τ=(1+5)/2≈1.618 \tau = (1+\sqrt{5})/2 \approx 1.618 τ=(1+5​)/2≈1.618; Fibonacci anyons have fusion rules τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ, directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Majorana fermions subject to the non-Abelian braid group are believed to be the basic ingredients of future topological quantum computations. In this work, we propose to simulate Majorana fermions of the Kitaev model in electric circuits. We generate an arbitral number of segments in a Kitaev chain which are in the topological phase. A pair of Majorana-like edge states emerge at the edges of a topological sector, which is observed by impedance peaks. When the Kitaev chain contains NNN topological sectors, the degeneracy of the zero-admittance states is 2N2^{N}2N. A set of AC voltages along the chain plays a role of the wave function. An \textit{LC} resonator attached to each Majorana-like edge state registers the phase and the absolute value of the voltage. One qubit consists of the in-phase and opposite-phase states made of two \textit{LC} resonators installed in one topological sector. It is possible to braid any pair of neighboring edge states with the aid of T-junction geometry. By calculating the Berry phase acquired by their eigenfunctions, the braiding is shown to generate one-qubit and two-qubit unitary operations. We explicitly construct Clifford quantum gates based on them. The present scheme may simulate topological quantum computations based on electric circuits.