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arXiv:2505.08888

Majorana Edge Modes as Quantum Memory for Topological Quantum Computing

Jasmin Bedow, Dirk K. Morr

Published May 13, 2025• 0 citations
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Abstract

We demonstrate that a combination of Majorana edge modes (MEMs) and Majorana zero modes (MZMs) located in the vortex cores of two-dimensional topological superconductors represent a new platform for the efficient implementation of fault-tolerant quantum gates. By calculating the full many-body dynamics of the system, we demonstrate the successful simulation and visualization of $Z$-, $X$- and Hadamard gates, with MEMs being functionalized as quantum memory. Our results open a new platform for the efficient implementation of fault-tolerant quantum computing.

DOI
10.48550/arxiv.2505.08888

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We demonstrate that a combination of Majorana edge modes (MEMs) and Majorana zero modes (MZMs) located in the vortex cores of two-dimensional topological superconductors represent a new platform for the efficient implementation of fault-tolerant quantum gates. By calculating the full many-body dynamics of the system, we demonstrate the successful simulation and visualization of ZZZ-, XXX- and Hadamard gates, with MEMs being functionalized as quantum memory. Our results open a new platform for the efficient implementation of fault-tolerant quantum computing.
The idea of topological quantum computation (TQC) is to store and manipulate quantum information in an intrinsically fault-tolerant manner by utilizing the physics of topologically ordered phases of matter. Currently, one of the most promising platforms for a topological qubit is in terms of Majorana fermion zero modes (MZMs) in spin-orbit coupled superconducting nanowires. However, the topologically robust operations that are possible with MZMs can be efficiently simulated on a classical computer and are therefore not sufficient for realizing a universal gate set for TQC. Here, we show that an array of coupled semiconductor-superconductor nanowires with MZM edge states can be used to realize a more sophisticated type of non-Abelian defect: a genon in an Ising ×\times× Ising topological state. This leads to a possible implementation of the missing topologically protected π/8π/8π/8 phase gate and thus universal TQC based on semiconductor-superconductor nanowire technology. We provide detailed numerical estimates of the relevant energy scales, which we show to lie within accessible ranges.
Executing quantum algorithms using Majorana zero modes—a major milestone for the field of topological quantum computing—requires a platform that can be scaled to large quantum registers, can be controlled in real time and space, and a braiding protocol that uses the unique properties of these exotic particles. Here, we demonstrate the first successful simulation of a Majorana-based, fault-tolerant quantum algorithm to solve the Bernstein-Vazirani problem in two-dimensional magnet-superconductor hybrid structures from initialization to read-out of the final many-body state. Utilizing the Majorana zero modes’ topological properties, we introduce an optimized braiding protocol for the algorithm and a scalable architecture for its implementation with an arbitrary number of qubits. We visualize the algorithm protocol in real time and space by computing the non-equilibrium density of states, which is proportional to the time-dependent differential conductance, and the non-equilibrium charge density, which assigns a unique signature to each final state of the algorithm.
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.
Topological states of matter are promising resources for composing fault-tolerant quantum computers, advancing beyond the limitations of current noisy intermediate-scale quantum devices. To enable this progress, a deep understanding of topological phenomena within actual quantum computing platforms is essential. However, existing quantum-circuit algorithms to examine topological properties remain limited. Here we introduce three quantum-circuit algorithms designed to (i) determine the ground state within a specified parity subspace, (ii) identify the many-body topological invariant, and (iii) visualize zero-energy edge modes. We illustrate these algorithms with the interacting Kitaev chain, a typical model of one-dimensional topological superconductors. These approaches are versatile, extending beyond one-dimensional systems to various topological states, including those in higher dimensions.
We propose and analyze a physical system capable of performing topological quantum computation with Majorana zero modes (MZMs) in a one-dimensional topological superconductor (1DTS). One of the leading methods to realize quantum gates in a 1DTS is to use T junctions, which allows one to maneuver MZMs in such a manner as to achieve braiding. In this paper, we propose a scheme for implementing quantum logical gates in a purely one-dimensional geometry without T junctions, instead replacing it with an auxiliary qubit. This has the additional benefit of introducing a non-Clifford gate, corresponding to one- and two-logical-qubit ZZZ rotations. We first design a topologically protected logical ZZZ gate based entirely on local interactions within the 1DTS. Using an auxiliary qubit coupled to the topological superconductors, we extend the ZZZ gate to non-Clifford single- and multiqubit arbitrary rotations with partial topological protection. Finally, to perform universal quantum computing, we introduce a scheme for performing arbitrary unitary rotations, albeit without topological protection. We develop a formalism based on unitary braids which creates transitions between different topological phases of the 1DTS system. The unitary formalism can be simply converted to an equivalent adiabatic scheme, which we numerically simulate, and we show that high-fidelity operations should be possible with reasonable parameters.