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:1305.1808 · Physical Review A

Topological phases and self-correcting memories in interacting anyon systems

James R. Wootton

Published Dec 10, 2013• 10 citations
arXivHTMLPDF

Abstract

Recent studies have shown that topological models with interacting anyonic quasiparticles can be used as self-correcting quantum memories. Here we study the behavior of these models at thermal equilibrium. It is found that the interactions allow topological order to exist at finite temperature, not only in an extension of the ground-state phase, but also in a novel form of topologically ordered phase. Both phases are found to support self-correction in all models considered, and the transition between them corresponds to a change in the scaling of memory lifetime with system size.

DOI
10.1103/physreva.88.062312

Related papers

Long-Range Toric Code: Finite-Temperature Topological Order in Two Dimensions

Jacopo Gliozzi, Taylor L. Hughes, Kyung-Su Kim

We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.

cond-mat.str-elCross-listedarXiv:2610.37765•Oct 29, 2026•0 citations

Self-correcting quantum memory in a thermal environment

Stefano Chesi, Beat Röthlisberger, Daniel Loss

The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection (``self-correction''), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose two-dimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive long-range interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive long-range nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that self-correcting quantum memories can exist in 2D at finite temperatures.

arXiv:0908.4264•Physical Review A•Aug 6, 2010•61 citations

Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems

Courtney G Brell, Simon Burton, Guillaume Dauphinais +2 more

We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath.We propose a phenomenological model for the resulting short-time dynamics that includes pair creation, hopping, braiding, and fusion of anyons.By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of the quantum information stored in the encoded spaces.To decode and correct errors in these codes, we adapt several existing topological decoders to the non-Abelian setting.We perform large-scale numerical simulations of these two-dimensional Ising anyon systems and find that the thresholds of these models range from 13% to 25%.To our knowledge, these are the first numerical threshold estimates for quantum codes without explicit additive structure.

arXiv:1311.0019•Physical Review X•Sep 30, 2014•41 citations

Thermalization of topological entropy after a quantum quench

Yu Zeng, Alioscia Hamma, Heng Fan

Topologically ordered quantum phases are robust in the sense that perturbations in the Hamiltonian of the system will not change the topological nature of the ground-state wave function. However, in order to exploit topological order for applications such as self-correcting quantum memories and information processing, these states need to be also robust both dynamically and at finite temperature in the presence of an environment. It is well known that systems like the toric code in two spatial dimensions are fragile in temperature. In this paper, we show a completely analytic treatment of the toric code away from equilibrium, after a quantum quench of the system Hamiltonian. We show that, despite being subject to unitary evolution (and at zero temperature), the long-time behavior of the topological entropy is thermal, therefore vanishing. If the quench preserves a local gauge structure, there is a residual long-lived topological entropy. This also is the thermal behavior in presence of such gauge constraints. The result is obtained by studying the time evolution of the topological 2-R\'enyi entropy in a fully analytical, exact way.

arXiv:1509.08613•Physical review. B./Physical review. B•Sep 2, 2016•12 citations

Toric-boson model: Toward a topological quantum memory at finite temperature

Alioscia Hamma, Claudio Castelnovo, Claudio Chamon

We discuss the existence of stable topological quantum memory at finite temperature. At stake here is the fundamental question of whether it is, in principle, possible to store quantum information for macroscopic times without the intervention from the external world, that is, without error correction. We study the toric code in two dimensions with an additional bosonic field that couples to the defects, in the presence of a generic environment at finite temperature: the toric-boson model. Although the coupling constants for the bare model are not finite in the thermodynamic limit, the model has a finite spectrum. We show that in the topological phase, there is a finite temperature below which open strings are confined and therefore the lifetime of the memory can be made arbitrarily (polynomially) long in system size. The interaction with the bosonic field yields a long-range attractive force between the end points of open strings but leaves closed strings and topological order intact.

arXiv:0812.4622•Physical Review B•Jun 18, 2009•67 citations

A witness for topological order and stable quantum memories in Abelian anyonic systems

James R. Wootton

We propose a novel parameter, the anyonic topological entropy, designed to detect the error correcting phase of a topological memory. Unlike similar quantities such as the topological entropy, the anyonic topological entropy is defined using the states of the anyon occupations. As such, though the parameter deals with phases and phase transitions that are quantum in nature, it can be calculated solely from classical probability distributions. In many cases, these calculations will be tractable using efficient classical algorithms. The parameter therefore provides a new avenue for efficient studies of anyonic systems.

arXiv:1112.2527•Journal of Physics A Mathematical and Theoretical•Sep 7, 2012•2 citations
Recent studies have shown that topological models with interacting anyonic quasiparticles can be used as self-correcting quantum memories. Here we study the behavior of these models at thermal equilibrium. It is found that the interactions allow topological order to exist at finite temperature, not only in an extension of the ground-state phase, but also in a novel form of topologically ordered phase. Both phases are found to support self-correction in all models considered, and the transition between them corresponds to a change in the scaling of memory lifetime with system size.
We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.
The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection (``self-correction''), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose two-dimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive long-range interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive long-range nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that self-correcting quantum memories can exist in 2D at finite temperatures.
We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath.We propose a phenomenological model for the resulting short-time dynamics that includes pair creation, hopping, braiding, and fusion of anyons.By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of the quantum information stored in the encoded spaces.To decode and correct errors in these codes, we adapt several existing topological decoders to the non-Abelian setting.We perform large-scale numerical simulations of these two-dimensional Ising anyon systems and find that the thresholds of these models range from 13% to 25%.To our knowledge, these are the first numerical threshold estimates for quantum codes without explicit additive structure.
Topologically ordered quantum phases are robust in the sense that perturbations in the Hamiltonian of the system will not change the topological nature of the ground-state wave function. However, in order to exploit topological order for applications such as self-correcting quantum memories and information processing, these states need to be also robust both dynamically and at finite temperature in the presence of an environment. It is well known that systems like the toric code in two spatial dimensions are fragile in temperature. In this paper, we show a completely analytic treatment of the toric code away from equilibrium, after a quantum quench of the system Hamiltonian. We show that, despite being subject to unitary evolution (and at zero temperature), the long-time behavior of the topological entropy is thermal, therefore vanishing. If the quench preserves a local gauge structure, there is a residual long-lived topological entropy. This also is the thermal behavior in presence of such gauge constraints. The result is obtained by studying the time evolution of the topological 2-R\'enyi entropy in a fully analytical, exact way.
We discuss the existence of stable topological quantum memory at finite temperature. At stake here is the fundamental question of whether it is, in principle, possible to store quantum information for macroscopic times without the intervention from the external world, that is, without error correction. We study the toric code in two dimensions with an additional bosonic field that couples to the defects, in the presence of a generic environment at finite temperature: the toric-boson model. Although the coupling constants for the bare model are not finite in the thermodynamic limit, the model has a finite spectrum. We show that in the topological phase, there is a finite temperature below which open strings are confined and therefore the lifetime of the memory can be made arbitrarily (polynomially) long in system size. The interaction with the bosonic field yields a long-range attractive force between the end points of open strings but leaves closed strings and topological order intact.
We propose a novel parameter, the anyonic topological entropy, designed to detect the error correcting phase of a topological memory. Unlike similar quantities such as the topological entropy, the anyonic topological entropy is defined using the states of the anyon occupations. As such, though the parameter deals with phases and phase transitions that are quantum in nature, it can be calculated solely from classical probability distributions. In many cases, these calculations will be tractable using efficient classical algorithms. The parameter therefore provides a new avenue for efficient studies of anyonic systems.