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:2305.18141 · Quantum

Entanglement dynamics in U(1) symmetric hybrid quantum automaton circuits

Yiqiu Han, Xiao Chen

Published Dec 6, 2023• 8 citations
arXivHTMLPDF

Abstract

We study the entanglement dynamics of quantum automaton (QA) circuits in the presence of U(1) symmetry. We find that the second Rényi entropy grows diffusively with a logarithmic correction as tln⁡t , saturating the bound established by Huang \cite{Huang_2020}. Thanks to the special feature of QA circuits, we understand the entanglement dynamics in terms of a classical bit string model. Specifically, we argue that the diffusive dynamics stems from the rare slow modes containing extensively long domains of spin 0s or 1s. Additionally, we investigate the entanglement dynamics of monitored QA circuits by introducing a composite measurement that preserves both the U(1) symmetry and properties of QA circuits. We find that as the measurement rate increases, there is a transition from a volume-law phase where the second Rényi entropy persists the diffusive growth (up to a logarithmic correction) to a critical phase where it grows logarithmically in time. This interesting phenomenon distinguishes QA circuits from non-automaton circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an area-law phase transition exists, and any non-zero rate of projective measurements in the volume-law phase leads to a ballistic growth of the Rényi entropy.

DOI
10.22331/q-2023-12-06-1200

Related papers

Measurement-induced criticality in Z2 -symmetric quantum automaton circuits

Yiqiu Han, Xiao Chen

We study entanglement dynamics in hybrid ${\mathbb{Z}}_{2}$-symmetric quantum automaton circuits subject to local composite measurements. We show that there exists an entanglement phase transition from a volume-law phase to a critical phase by varying the measurement rate $p$. By analyzing the underlying classical bit-string dynamics, we demonstrate that the critical point belongs to parity-conserving universality class. We further show that the critical phase with $p>{p}_{c}$ is related to the diffusion-annihilation process and is protected by the ${\mathbb{Z}}_{2}$-symmetric measurement. We give an interpretation of the entanglement entropy in terms of a two-species particle model and identify the coefficient in front of the critical logarithmic entanglement scaling as the local persistent coefficient. The critical behavior observed at $p\ensuremath{\ge}{p}_{c}$ and the associated dynamical exponents are also confirmed in the purification dynamics.

arXiv:2110.10726•Physical review. B./Physical review. B•Feb 28, 2022•23 citations

Entanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits

Utkarsh Agrawal, Aidan Zabalo, Kun Chen +5 more

Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements.We study how entanglement dynamics in nonunitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers.We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system.We find that while Rényi entropies grow sub-ballistically as ffiffi t p in the absence of measurement, for even an infinitesimal rate of measurements, all average Rényi entropies grow ballistically with time ∼t.We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes.Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively stabilized symmetry-breaking and topological orders.

arXiv:2107.10279•Physical Review X•Oct 7, 2022•137 citations

Entanglement structure in the volume-law phase of hybrid quantum automaton circuits

Yiqiu Han, Xiao Chen

We study entanglement fluctuations and quantum error correction in the weakly monitored volume-law phase of quantum automaton circuits subject to repeated local measurements. We numerically observe that the entanglement entropy exhibits strong fluctuation with the exponent close to the ``growth exponent'' of the Kardar-Parisi-Zhang (KPZ) universality class, the same as other local random circuits studied previously. We also investigate the dynamically generated quantum error correction code in the purification process and show that this model has different contiguous code distances for two types of errors that exhibit similar sublinear power-law scaling. We give an interpretation of these results by mapping them to various quantities in a classical particle model. We demonstrate that the subleading correction term of the entanglement entropy and the sublinear power-law scaling of the contiguous code distance in the volume-law phase are both the emergent phenomena of the hybrid random dynamics. Finally, we show that this classical particle dynamics itself has a type of error correction ability and can dynamically generate a classical linear code.

arXiv:2207.02165•Physical review. B./Physical review. B•Jan 18, 2023•7 citations

From Quantum Circuits with Ultraslow Dynamics to Classical Plaquette Models

Vikram Ravindranath, Hanchen Liu, Xiao Chen

We introduce a family of hybrid quantum circuits involving unitary gates and projective measurements that display a measurement-induced phase transition. Remarkably, the volume-law phase featuring logarithmic entanglement growth for certain initial states. We attribute this slow entanglement growth to the similarly slow growth of the participation entropy, which bounds the entanglement. Furthermore, the quantum circuit can be mapped to a classical spin model with real positive Boltzmann weights which involves local multi-spin interactions and displays glassy dynamics at finite temperature. We trace the origin of both the slow quantum dynamics and the classical glassiness to the presence of large, non-local symmetry operators. Our work establishes a novel connection between quantum entanglement dynamics and classical glassy behavior, offering a new geometric perspective on entanglement phase transitions.

arXiv:2510.07247•Oct 8, 2025•0 citations

Classical Models of Entanglement in Monitored Random Circuits

Oles Shtanko, Yaroslav A. Kharkov, Luis Pedro García-Pintos +1 more

The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth Rényi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-Rényi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.

arXiv:2004.06736•Apr 14, 2020•24 citations

Sub-ballistic Growth of Rényi Entropies due to Diffusion

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as ∝sqrt[t], in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.

arXiv:1901.10502•Physical Review Letters•Jun 25, 2019•144 citations
We study the entanglement dynamics of quantum automaton (QA) circuits in the presence of U(1) symmetry. We find that the second Rényi entropy grows diffusively with a logarithmic correction as tln⁡t , saturating the bound established by Huang \cite{Huang_2020}. Thanks to the special feature of QA circuits, we understand the entanglement dynamics in terms of a classical bit string model. Specifically, we argue that the diffusive dynamics stems from the rare slow modes containing extensively long domains of spin 0s or 1s. Additionally, we investigate the entanglement dynamics of monitored QA circuits by introducing a composite measurement that preserves both the U(1) symmetry and properties of QA circuits. We find that as the measurement rate increases, there is a transition from a volume-law phase where the second Rényi entropy persists the diffusive growth (up to a logarithmic correction) to a critical phase where it grows logarithmically in time. This interesting phenomenon distinguishes QA circuits from non-automaton circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an area-law phase transition exists, and any non-zero rate of projective measurements in the volume-law phase leads to a ballistic growth of the Rényi entropy.
We study entanglement dynamics in hybrid Z2{\mathbb{Z}}_{2}Z2​-symmetric quantum automaton circuits subject to local composite measurements. We show that there exists an entanglement phase transition from a volume-law phase to a critical phase by varying the measurement rate ppp. By analyzing the underlying classical bit-string dynamics, we demonstrate that the critical point belongs to parity-conserving universality class. We further show that the critical phase with p>pcp>{p}_{c}p>pc​ is related to the diffusion-annihilation process and is protected by the Z2{\mathbb{Z}}_{2}Z2​-symmetric measurement. We give an interpretation of the entanglement entropy in terms of a two-species particle model and identify the coefficient in front of the critical logarithmic entanglement scaling as the local persistent coefficient. The critical behavior observed at $p\ensuremath{\ge}{p}_{c}$ and the associated dynamical exponents are also confirmed in the purification dynamics.
Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements.We study how entanglement dynamics in nonunitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers.We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system.We find that while Rényi entropies grow sub-ballistically as ffiffi t p in the absence of measurement, for even an infinitesimal rate of measurements, all average Rényi entropies grow ballistically with time ∼t.We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes.Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively stabilized symmetry-breaking and topological orders.
We study entanglement fluctuations and quantum error correction in the weakly monitored volume-law phase of quantum automaton circuits subject to repeated local measurements. We numerically observe that the entanglement entropy exhibits strong fluctuation with the exponent close to the ``growth exponent'' of the Kardar-Parisi-Zhang (KPZ) universality class, the same as other local random circuits studied previously. We also investigate the dynamically generated quantum error correction code in the purification process and show that this model has different contiguous code distances for two types of errors that exhibit similar sublinear power-law scaling. We give an interpretation of these results by mapping them to various quantities in a classical particle model. We demonstrate that the subleading correction term of the entanglement entropy and the sublinear power-law scaling of the contiguous code distance in the volume-law phase are both the emergent phenomena of the hybrid random dynamics. Finally, we show that this classical particle dynamics itself has a type of error correction ability and can dynamically generate a classical linear code.
We introduce a family of hybrid quantum circuits involving unitary gates and projective measurements that display a measurement-induced phase transition. Remarkably, the volume-law phase featuring logarithmic entanglement growth for certain initial states. We attribute this slow entanglement growth to the similarly slow growth of the participation entropy, which bounds the entanglement. Furthermore, the quantum circuit can be mapped to a classical spin model with real positive Boltzmann weights which involves local multi-spin interactions and displays glassy dynamics at finite temperature. We trace the origin of both the slow quantum dynamics and the classical glassiness to the presence of large, non-local symmetry operators. Our work establishes a novel connection between quantum entanglement dynamics and classical glassy behavior, offering a new geometric perspective on entanglement phase transitions.
The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth Rényi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-Rényi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.
We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as ∝sqrt[t], in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.