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:2008.02816 · Physical Review Letters

Symmetry Breaking and Error Correction in Open Quantum Systems

Simon Lieu, Ron Belyansky, Jeremy T. Young, Rex Lundgren, Victor V. Albert, Alexey Vyacheslavovich Gorshkov

Published Dec 8, 2020• 116 citations
arXivHTMLPDF

Abstract

Symmetry-breaking transitions are a well-understood phenomenon of closed quantum systems in quantum optics, condensed matter, and high energy physics. However, symmetry breaking in open systems is less thoroughly understood, in part due to the richer steady-state and symmetry structure that such systems possess. For the prototypical open system-a Lindbladian-a unitary symmetry can be imposed in a "weak" or a "strong" way. We characterize the possible Z_{n} symmetry-breaking transitions for both cases. In the case of Z_{2}, a weak-symmetry-broken phase guarantees at most a classical bit steady-state structure, while a strong-symmetry-broken phase admits a partially protected steady-state qubit. Viewing photonic cat qubits through the lens of strong-symmetry breaking, we show how to dynamically recover the logical information after any gap-preserving strong-symmetric error; such recovery becomes perfect exponentially quickly in the number of photons. Our study forges a connection between driven-dissipative phase transitions and error correction.

DOI
10.1103/physrevlett.125.240405

Related papers

Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking

Jeet Shah, Christopher Fechisin, Yu-Xin Wang +6 more

Recent experimental progress in controlling open quantum systems enables the pursuit of mixed-state nonequilibrium quantum phases. We investigate whether open quantum systems hosting mixed-state symmetry-protected topological states as steady states retain this property under symmetric perturbations. Focusing on the decohered cluster state – a mixed-state symmetry-protected topological state protected by a combined strong and weak symmetry – we construct a parent Lindbladian that hosts it as a steady state. This Lindbladian can be mapped onto exactly solvable reaction-diffusion dynamics, even in the presence of certain perturbations, allowing us to solve the parent Lindbladian in detail and reveal previously-unknown steady states. Using both analytical and numerical methods, we find that typical symmetric perturbations cause strong-to-weak spontaneous symmetry breaking at arbitrarily small perturbations, destabilize the steady-state mixed-state symmetry-protected topological order. However, when perturbations introduce only weak symmetry defects, the steady-state mixed-state symmetry-protected topological order remains stable. Additionally, we construct a quantum channel which replicates the essential physics of the Lindbladian and can be efficiently simulated using only Clifford gates, Pauli measurements, and feedback.

arXiv:2410.12900•Quantum•Nov 17, 2025•5 citations

Information dynamics and symmetry breaking in generic monitored $\mathbb{Z}_2$-symmetric open quantum systems

Jacob Hauser, Ali Lavasani, Sagar Vijay +1 more

We investigate the steady-state phases of generic $\mathbb{Z}_2$-symmetric monitored, open quantum dynamics. We describe the phases systematically in terms of both information-theoretic diagnostics and spontaneous breaking of strong and weak symmetries of the dynamics. We find a completely broken phase where information is retained by the quantum system, a strong-to-weak broken phase where information is leaked to the environment, and an unbroken phase where information is learned by the observer. We find that weak measurement and dephasing alone constitute a minimal model for generic open systems with $\mathbb{Z}_2$ symmetry, but we also explore perturbations by unitary gates. For a 1d set of qubits, we examine information-theoretic and symmetry-breaking observables in the path integral of the doubled state. This path integral reduces to the standard classical 2d random-bond Ising model in certain limits but generically involves negative weights, enabling a special self-dual random-bond Ising model at the critical point when only measurements are present. We obtain numerical evidence for the steady-state phases using efficient tensor network simulations of the doubled state.

quant-pharXiv:2512.03031•Dec 3, 2025•0 citations

Harnessing strong symmetry breaking finite-component system phase transition for quantum error correction

Yanzhang Zhu, Myung-Joong Hwang

We investigate a dissipative finite-component system phase transition induced by two photon loss mechanism, which features a strong symmetry breaking of the Liouvillian. We characterize the mean-field phase diagram and demonstrate that the system could be harnessed to implement passive error correction for a hybrid qubit-oscillator qubit.

Quantum 2.0 Conference and Exhibition•Jan 1, 2024•0 citations

Dissipative phase transitions and passive error correction

Yu-Jie Liu, Simon Lieu

We classify different ways to passively protect classical and quantum information, i.e., we do not allow for syndrome measurements, in the context of local Lindblad models for spin systems. Within this family of models, we suggest that passive error correction is associated with nontrivial phases of matter and propose a definition for dissipative phases based on robust steady-state degeneracy of a Lindbladian in the thermodynamic limit. We study three thermalizing models in this context: the two-dimensional (2D) Ising model, the 2D toric code, and the 4D toric code. In the low-temperature phase, the 2D Ising model hosts a robust classical steady-state degeneracy, while the 4D toric code hosts a robust quantum steady-state degeneracy. We perturb the models with terms that violate detailed balance and observe that qualitative features remain unchanged, suggesting that ${\mathbb{Z}}_{2}$ symmetry breaking in a Lindbladian is useful to protect a classical bit while intrinsic topological order protects a qubit.

arXiv:2307.09512•Physical Review A•Feb 16, 2024•25 citations

Emergence of PT-symmetry breaking in open quantum systems

Julian Huber, Peter Kirton, Stefan Rotter +1 more

The effect of \mathcal{PT} 𝒫 𝒯 -symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT} 𝒫 𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT} 𝒫 𝒯 -symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT} 𝒫 𝒯 -symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.

arXiv:2003.02265•SciPost Physics•Oct 19, 2020•64 citations

Edge Modes and Symmetry-Protected Topological States in Open Quantum Systems

Dawid Paszko, Dominic C. Rose, Marzena Hanna Szymańska +1 more

Topological order offers possibilities for processing quantum information that can be immune to imperfections. However, the question of its stability out of equilibrium is relevant for experiments, where coupling to an environment is unavoidable. In this work, we demonstrate the robustness of certain aspects of Z 2 × Z 2 symmetry-protected topological (SPT) order against a wide class of dissipation channels in the Lindblad and quantum trajectory formalisms of an open quantum system. This is illustrated using the one-dimensional Z X Z cluster Hamiltonian along with Pauli-string jump operators. We show that certain choices of dissipation retaining strong symmetries support a steady-state manifold consisting of two nonlocal qubits and for Hamiltonian perturbations preserving the global symmetry, states in this manifold remain metastable. In contrast, this metastability is destroyed upon breaking the above-mentioned symmetry. While the localized qubits of the cluster Hamiltonian are not conserved by the Lindbladian evolution, they do correspond to weak symmetries and thus retain a memory of their initial state at all times in the quantum trajectories. We utilize this feature to construct protocols to retrieve the quantum information either by monitoring jumps or error mitigation. Our work thus proposes a novel framework to study the dynamics of dissipative SPT phases and opens up the possibility of engineering entangled states relevant to quantum information processing. Published by the American Physical Society 2024

arXiv:2310.09406•PRX Quantum•Jul 8, 2024•15 citations
Symmetry-breaking transitions are a well-understood phenomenon of closed quantum systems in quantum optics, condensed matter, and high energy physics. However, symmetry breaking in open systems is less thoroughly understood, in part due to the richer steady-state and symmetry structure that such systems possess. For the prototypical open system-a Lindbladian-a unitary symmetry can be imposed in a "weak" or a "strong" way. We characterize the possible Z_{n} symmetry-breaking transitions for both cases. In the case of Z_{2}, a weak-symmetry-broken phase guarantees at most a classical bit steady-state structure, while a strong-symmetry-broken phase admits a partially protected steady-state qubit. Viewing photonic cat qubits through the lens of strong-symmetry breaking, we show how to dynamically recover the logical information after any gap-preserving strong-symmetric error; such recovery becomes perfect exponentially quickly in the number of photons. Our study forges a connection between driven-dissipative phase transitions and error correction.
Recent experimental progress in controlling open quantum systems enables the pursuit of mixed-state nonequilibrium quantum phases. We investigate whether open quantum systems hosting mixed-state symmetry-protected topological states as steady states retain this property under symmetric perturbations. Focusing on the decohered cluster state – a mixed-state symmetry-protected topological state protected by a combined strong and weak symmetry – we construct a parent Lindbladian that hosts it as a steady state. This Lindbladian can be mapped onto exactly solvable reaction-diffusion dynamics, even in the presence of certain perturbations, allowing us to solve the parent Lindbladian in detail and reveal previously-unknown steady states. Using both analytical and numerical methods, we find that typical symmetric perturbations cause strong-to-weak spontaneous symmetry breaking at arbitrarily small perturbations, destabilize the steady-state mixed-state symmetry-protected topological order. However, when perturbations introduce only weak symmetry defects, the steady-state mixed-state symmetry-protected topological order remains stable. Additionally, we construct a quantum channel which replicates the essential physics of the Lindbladian and can be efficiently simulated using only Clifford gates, Pauli measurements, and feedback.
Information dynamics and symmetry breaking in generic monitored Z2\mathbb{Z}_2Z2​-symmetric open quantum systems
We investigate the steady-state phases of generic Z2\mathbb{Z}_2Z2​-symmetric monitored, open quantum dynamics. We describe the phases systematically in terms of both information-theoretic diagnostics and spontaneous breaking of strong and weak symmetries of the dynamics. We find a completely broken phase where information is retained by the quantum system, a strong-to-weak broken phase where information is leaked to the environment, and an unbroken phase where information is learned by the observer. We find that weak measurement and dephasing alone constitute a minimal model for generic open systems with Z2\mathbb{Z}_2Z2​ symmetry, but we also explore perturbations by unitary gates. For a 1d set of qubits, we examine information-theoretic and symmetry-breaking observables in the path integral of the doubled state. This path integral reduces to the standard classical 2d random-bond Ising model in certain limits but generically involves negative weights, enabling a special self-dual random-bond Ising model at the critical point when only measurements are present. We obtain numerical evidence for the steady-state phases using efficient tensor network simulations of the doubled state.
We classify different ways to passively protect classical and quantum information, i.e., we do not allow for syndrome measurements, in the context of local Lindblad models for spin systems. Within this family of models, we suggest that passive error correction is associated with nontrivial phases of matter and propose a definition for dissipative phases based on robust steady-state degeneracy of a Lindbladian in the thermodynamic limit. We study three thermalizing models in this context: the two-dimensional (2D) Ising model, the 2D toric code, and the 4D toric code. In the low-temperature phase, the 2D Ising model hosts a robust classical steady-state degeneracy, while the 4D toric code hosts a robust quantum steady-state degeneracy. We perturb the models with terms that violate detailed balance and observe that qualitative features remain unchanged, suggesting that Z2{\mathbb{Z}}_{2}Z2​ symmetry breaking in a Lindbladian is useful to protect a classical bit while intrinsic topological order protects a qubit.
The effect of \mathcal{PT} 𝒫 𝒯 -symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT} 𝒫 𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT} 𝒫 𝒯 -symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT} 𝒫 𝒯 -symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.
Topological order offers possibilities for processing quantum information that can be immune to imperfections. However, the question of its stability out of equilibrium is relevant for experiments, where coupling to an environment is unavoidable. In this work, we demonstrate the robustness of certain aspects of Z 2 × Z 2 symmetry-protected topological (SPT) order against a wide class of dissipation channels in the Lindblad and quantum trajectory formalisms of an open quantum system. This is illustrated using the one-dimensional Z X Z cluster Hamiltonian along with Pauli-string jump operators. We show that certain choices of dissipation retaining strong symmetries support a steady-state manifold consisting of two nonlocal qubits and for Hamiltonian perturbations preserving the global symmetry, states in this manifold remain metastable. In contrast, this metastability is destroyed upon breaking the above-mentioned symmetry. While the localized qubits of the cluster Hamiltonian are not conserved by the Lindbladian evolution, they do correspond to weak symmetries and thus retain a memory of their initial state at all times in the quantum trajectories. We utilize this feature to construct protocols to retrieve the quantum information either by monitoring jumps or error mitigation. Our work thus proposes a novel framework to study the dynamics of dissipative SPT phases and opens up the possibility of engineering entangled states relevant to quantum information processing. Published by the American Physical Society 2024