Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1004.5533 · Physical Review B

Topological quantum computation away from the ground state using Majorana fermions

Anton Akhmerov

Published Jul 20, 2010• 106 citations
arXivHTMLPDF

Abstract

We relax one of the requirements for topological quantum computation with Majorana fermions. Topological quantum computation was discussed so far as manipulation of the wave function within degenerate many-body ground state. The simplest particles providing degenerate ground state, Majorana fermions, often coexist with extremely low-energy excitations so keeping the system in the ground state may be hard. We show that the topological protection extends to the excited states, as long as the Majorana fermions do not interact neither directly nor via the excited states. This protection relies on the fermion parity conservation and so it is generic to any implementation of Majorana fermions.

DOI
10.1103/physrevb.82.020509

Related papers

Exact zero modes and decoherence in systems of interacting Majorana fermions

Guang Yang, Dima E. Feldman

Majorana fermions often coexist with other low-energy fermionic degrees of freedom. In such a situation, topological quantum computation requires the use of fermionic zero modes of a many-body system. We classify all such modes for interacting fermions and show how to select the mode that maximizes the decoherence time. We find that in a typical interacting system the maximal decoherence time is within one order of magnitude from the decoherence time of a qubit based on the local part of the fermion parity operator.

arXiv:1311.7604•Physical Review B•Jan 22, 2014•14 citations

Systematic construction of topological-nontopological hybrid universal quantum gates based on many-body Majorana fermion interactions

Motohiko Ezawa

Topological quantum computation by way of braiding of Majorana fermions is not universal quantum computation. There are several attempts to make universal quantum computation by introducing some additional quantum gates or quantum states. However, there is a serious problem in embedding an $M$-qubit quantum gate in the $N$-qubit system with $N>M$. This is an inherent problem to the Majorana system, where quantum gates for logical qubits become nonlocal in terms of physical qubits in general because braiding operations preserve the fermion parity. For instance, the CZ gate could not be embedded in the three-qubit system. We overcome this embedding problem by introducing $2(N+1)$-body interactions of Majorana fermions in the $N$-qubit system. A universal set of quantum gates is constructed for $N$ logical qubits, leading to topological-nontopological hybrid universal quantum computation. It would be more robust than conventional universal quantum computation because the quantum gates generated by braiding are topologically protected.

arXiv:2304.06260•Physical review. B./Physical review. B•Jul 9, 2024•5 citations

Local symmetry enhanced Majorana zero modes in one-dimensional fermionic systems with particle-number preserving interactions

Xing-Shuo Xu, Xiang-Fa Zhou, Guang‐Can Guo +1 more

The realization and manipulation of topological Majorana zero modes (MZMs) have been pivotal in both theoretical and experimental research. In this paper, we present a symmetry enhanced topological model that supports stable, localized MZMs in a one-dimensional interacting fermionic system. The MZMs in our model, induced by particle-number conserving interactions, are fully decoupled from the chain's interior and remain degenerate regardless of chain length, allowing for simulations in extremely short fermionic chains. The topological phase transition is driven by changes in coupling parameters, transitioning to a trivial phase without affecting the MZMs' independence or degeneracy, only their topological protection. Additionally, local symmetries in the model enhance MZMs' stability and reveal unique odd-even effects across different chain lengths.

Physical review. B./Physical review. B•Dec 19, 2024•0 citations

Topological Quantum Computation—From Basic Concepts to First Experiments

Ady Stern, Netanel H. Lindner

Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. Topological quantum computation aims to achieve this goal by using non-Abelian quantum phases of matter. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections in the implemented protocols and from interactions with the environment. Recently, substantial progress in this field has been made on both theoretical and experimental fronts. We review the basic concepts of non-Abelian phases and their topologically protected use in quantum information processing tasks. We discuss different possible realizations of these concepts in experimentally available solid-state systems, including systems hosting Majorana fermions, their recently proposed fractional counterparts, and non-Abelian quantum Hall states.

Science•Mar 7, 2013•339 citations

Majorana braiding in superconductors with fixed total number of particles

Ivar Martin, Kartiek Agarwal

One-dimensional topological superconductors treated at the mean-field level host zero-energy edge Majorana modes, which encode topological degeneracy of their ground states. Geometric manipulations (braiding) of multiple wires can be used to induce topologically robust transformations within the ground state subspace. The mean-field ground states do not have a definite number of particles and thus cannot describe an isolated system. Projecting such states onto fixed particle number gives a very good approximation to the true ground state of an isolated superconductor. In previous work, we showed that so projecting the prototypical Kitaev wave function of a single wire retains key features of the mean-field description, such as the zero-energy single-particle spectral peaks near the wire edges. Here we consider the case of multiple wires with conserved total charge. Again, using the projected Kitaev wave functions as the template, we identify the many-body counterparts of the mean-field topologically degenerate ground states. We then demonstrate how the quantum gates equivalent to braiding of Majorana zero modes can be implemented, albeit with reduced protection from noise and disorder.

arXiv:2407.12958•Jul 17, 2024•0 citations

Topological protection of Majorana qubits

Meng Cheng, Roman M. Lutchyn, S. Das Sarma

We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless ${p}_{x}+i{p}_{y}$ superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.

arXiv:1112.3662•Physical Review B•Apr 16, 2012•87 citations
We relax one of the requirements for topological quantum computation with Majorana fermions. Topological quantum computation was discussed so far as manipulation of the wave function within degenerate many-body ground state. The simplest particles providing degenerate ground state, Majorana fermions, often coexist with extremely low-energy excitations so keeping the system in the ground state may be hard. We show that the topological protection extends to the excited states, as long as the Majorana fermions do not interact neither directly nor via the excited states. This protection relies on the fermion parity conservation and so it is generic to any implementation of Majorana fermions.
Majorana fermions often coexist with other low-energy fermionic degrees of freedom. In such a situation, topological quantum computation requires the use of fermionic zero modes of a many-body system. We classify all such modes for interacting fermions and show how to select the mode that maximizes the decoherence time. We find that in a typical interacting system the maximal decoherence time is within one order of magnitude from the decoherence time of a qubit based on the local part of the fermion parity operator.
Topological quantum computation by way of braiding of Majorana fermions is not universal quantum computation. There are several attempts to make universal quantum computation by introducing some additional quantum gates or quantum states. However, there is a serious problem in embedding an MMM-qubit quantum gate in the NNN-qubit system with N>MN>MN>M. This is an inherent problem to the Majorana system, where quantum gates for logical qubits become nonlocal in terms of physical qubits in general because braiding operations preserve the fermion parity. For instance, the CZ gate could not be embedded in the three-qubit system. We overcome this embedding problem by introducing 2(N+1)2(N+1)2(N+1)-body interactions of Majorana fermions in the NNN-qubit system. A universal set of quantum gates is constructed for NNN logical qubits, leading to topological-nontopological hybrid universal quantum computation. It would be more robust than conventional universal quantum computation because the quantum gates generated by braiding are topologically protected.
The realization and manipulation of topological Majorana zero modes (MZMs) have been pivotal in both theoretical and experimental research. In this paper, we present a symmetry enhanced topological model that supports stable, localized MZMs in a one-dimensional interacting fermionic system. The MZMs in our model, induced by particle-number conserving interactions, are fully decoupled from the chain's interior and remain degenerate regardless of chain length, allowing for simulations in extremely short fermionic chains. The topological phase transition is driven by changes in coupling parameters, transitioning to a trivial phase without affecting the MZMs' independence or degeneracy, only their topological protection. Additionally, local symmetries in the model enhance MZMs' stability and reveal unique odd-even effects across different chain lengths.
Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. Topological quantum computation aims to achieve this goal by using non-Abelian quantum phases of matter. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections in the implemented protocols and from interactions with the environment. Recently, substantial progress in this field has been made on both theoretical and experimental fronts. We review the basic concepts of non-Abelian phases and their topologically protected use in quantum information processing tasks. We discuss different possible realizations of these concepts in experimentally available solid-state systems, including systems hosting Majorana fermions, their recently proposed fractional counterparts, and non-Abelian quantum Hall states.
One-dimensional topological superconductors treated at the mean-field level host zero-energy edge Majorana modes, which encode topological degeneracy of their ground states. Geometric manipulations (braiding) of multiple wires can be used to induce topologically robust transformations within the ground state subspace. The mean-field ground states do not have a definite number of particles and thus cannot describe an isolated system. Projecting such states onto fixed particle number gives a very good approximation to the true ground state of an isolated superconductor. In previous work, we showed that so projecting the prototypical Kitaev wave function of a single wire retains key features of the mean-field description, such as the zero-energy single-particle spectral peaks near the wire edges. Here we consider the case of multiple wires with conserved total charge. Again, using the projected Kitaev wave functions as the template, we identify the many-body counterparts of the mean-field topologically degenerate ground states. We then demonstrate how the quantum gates equivalent to braiding of Majorana zero modes can be implemented, albeit with reduced protection from noise and disorder.
We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless px+ipy{p}_{x}+i{p}_{y}px​+ipy​ superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.