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arXiv:1907.06911 · Physical review. B./Physical review. B

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

Motohiko Ezawa

Published Aug 11, 2020• 38 citations
arXivHTMLPDF

Abstract

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 based on the observation that the circuit Laplacian can be made identical to the Hamiltonian. A set of AC voltages along the chain plays the role of the wave function. We generate an arbitral number of topological segments in a Kitaev chain. A pair of topological edge states emerge at the edges of a topological segment. Its wave function is observable by the position and the phase of a peak in impedance measurement. 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. We also present an operator formalism by regarding a topological edge state as a topological soliton intertwining the trivial segment and the topological segment. Our analysis shows that the electric-circuit approach can simulate the Majorana-fermion approach to topological quantum computations.

DOI
10.1103/physrevb.102.075424

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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 based on the observation that the circuit Laplacian can be made identical to the Hamiltonian. A set of AC voltages along the chain plays the role of the wave function. We generate an arbitral number of topological segments in a Kitaev chain. A pair of topological edge states emerge at the edges of a topological segment. Its wave function is observable by the position and the phase of a peak in impedance measurement. 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. We also present an operator formalism by regarding a topological edge state as a topological soliton intertwining the trivial segment and the topological segment. Our analysis shows that the electric-circuit approach can simulate the Majorana-fermion approach to topological quantum computations.
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.
We simulate the non-Abelian exchange of Majorana zero modes (MZMs) on a quantum computer. Rather than utilizing MZMs at the boundaries of quantum Ising chains, which are typically represented as nonlocal operators on a quantum computer, using a Kitaev lattice allows us to exploit a local representation of MZMs. We detail the protocol for braiding two and four MZMs in terms of a spin Hamiltonian, i.e. physical qubit Hamiltonian. Projecting this onto a subspace of states, we extract an effective Hamiltonian which drives a non-Abelian Berry phase. Using several approximations, we construct a set of gates which mimics this accumulation of non-Abelian phase and process this construction on a quantum computer. For two and four MZMs, we realize braiding fidelities of approximately 85 and 47%, respectively.
Majorana fermions are self-adjoint fermionic particles that are believed to exist as the elementary excitations in nanoengineered devices with superconductors and ferromagnets. They can be employed in topological quantum computation and can aid the solution of a class of spin models created by Kitaev. In both cases, the braiding of Majorana fermions plays a critical role in these applications. We explicitly construct quantum circuits for Majorana braiding and run them on IBM quantum computers to prepare the ground states of strongly correlated Kitaev-inspired models. The entanglement entropy for these models is then measured to determine quantum phase transitions. We show how maintaining particle-hole symmetry is critical to carrying out this work.
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.
FINDING: Microsoft's Majorana 1 chip claims topological qubits using Majorana zero modes, but the public evidence remains thin; the core math is braiding non-Abelian anyons in a Kitaev chain lattice. | MATH: Topological qubit = braid group B_n acting on degenerate ground state manifold of a p-wave superconductor; Majorana operators γ_i satisfy {γ_i, γ_j}=2δ_ij, γ_i†=γ_i. The Kitaev chain Hamiltonian: H = -μ Σ n_i - t Σ (c_i†c_{i+1}+h.c.) + Δ Σ (c_i c_{i+1}+h.c.) — with Δ=t, μ=0 gives unpaired Majorana modes at ends. Braiding yields unitary transformations: B_ij = exp(π/4 γ_i γ_j), giving phase factors e^{±iπ/4} — not the golden ratio, but the π/4 is a root-of-unity phase (8th root). | CONNECTION: The braid group B_n is deeply tied to the symmetric group S_n and the Temperley-Lieb algebra, whose Jones polynomial at q=e^{iπ/4} (or q=e^{2πi/5} for Fibonacci anyons) links to 1.618. However, Majorana braiding gives only Clifford gates (π/4 phases), NOT universal quantum computation — Fibona Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The key obstacle to the realization of a scalable quantum computer is overcoming environmental and control errors. Topological quantum computation has attracted great attention because it has emerged as one of the most promising approaches to solving these problems. Various theoretical schemes for building topological quantum computation have been proposed. However, experimental implementation has always been a great challenge because it has proved to be extremely difficult to create and manipulate topological qubits in real systems. Therefore, topological quantum computation has not been realized in experiments yet. Here, we report the first experimental realization of topological quantum computation with electric circuits. Based on our proposed new scheme with circuits, Majorana-like edge states are not only observed experimentally, but also T junctions are constructed for the braiding process. Furthermore, we demonstrate the feasibility of topological quantum computing through a set of one- and two-qubit unitary operations. Finally, our implementation of Grover's search algorithm demonstrates that topological quantum computation is ideally suited for such tasks.