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:2405.00416 · Physical Review A

Entanglement and fidelity across quantum phase transitions in locally perturbed topological codes with open boundaries

K J Harikrishnan, Amit Kumar Pal

Published Mar 3, 2025• 2 citations
arXivHTMLPDF

Abstract

We investigate the topological-to-nontopological quantum phase transitions (QPTs) occurring in the Kitaev code under local perturbations in the form of a local magnetic field and spin-spin interactions of the Ising-type using fidelity susceptibility (FS) and entanglement as the probes. We assume the code to be embedded on the surface of a wide cylinder of height $M$ and circumference $D$ with $M\ensuremath{\ll}D$. We demonstrate a power-law divergence of FS across the QPT and determine the quantum critical points via a finite-size-scaling analysis. We verify these results by mapping the perturbed Kitaev code to the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions and computing the single-site magnetization as the order parameter using the quantum Monte Carlo technique. We also demonstrate a finite-size odd-even dichotomy in the occurrence of the QPT in the Kitaev ladder with respect to the odd and even values of $D$, when the system is perturbed with only Ising interaction. Our results also indicate a higher robustness of the topological phase of the Kitaev code against local perturbations if the boundary is made open along one direction. We further consider a local entanglement witness operator designed specifically to capture a lower bound to the localizable entanglement on the vertical nontrivial loop of the code. We show that the first derivative of the expectation value of the witness operator exhibits a logarithmic divergence across the QPT and perform the finite-size-scaling analysis. We demonstrate similar behavior of the expectation value of the appropriately constructed witness operator also in the case of a locally perturbed color code with open boundaries.

DOI
10.1103/physreva.111.032401

Related papers

Singularities in ground-state fidelity and quantum phase transitions for the Kitaev model

Jian-Hui Zhao, Huan-Qiang Zhou

The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.

arXiv:0803.0814•Physical Review B•Jul 1, 2009•58 citations

The toric code under antiferromagnetic isotropic Heisenberg interactions

Won Jang, Robert Peters, Thore Posske

We investigate the impact of an isotropic antiferromagnetic Heisenberg perturbation on the toric code, focusing on the resulting quantum phase transition and the nature of the phase that emerges beyond topological order. Using neural-network quantum states (NQS), we compute ground states over a wide range of Heisenberg couplings while fully respecting the exact symmetries of the model. In the weak-coupling regime, the numerical results are in excellent agreement with an effective low-energy description derived from a Schrieffer-Wolff (SW) transformation, providing analytic control over the perturbative breakdown of topological order. We show that the Heisenberg perturbation only renormalizes local operators at low orders, whereas mixing between topological sectors occurs only at a perturbative order proportional to the system size. At intermediate values of the Heisenberg interaction, the topological phase breaks down. We estimate the critical point through a combination of the fidelity susceptibility and the logarithmic susceptibility of non-contractible Wilson loops for various system sizes. Furthermore, we utilize the topological entanglement entropy to provide a comprehensive characterization of the phase transition. Beyond the transition, an antiferromagnetic $\pm X/\pm Z$ Néel phase emerges, characterized by a fourfold-degenerate symmetry-broken manifold, which is explicitly probed using staggered-magnetization-based diagnostics. Our results show how local two-spin interactions, which naturally arise in realistic implementations of the toric code, drive the breakdown of topological order. Moreover, we establish the SW approach as a systematic framework for analyzing such perturbations in combination with variational many-body methods.

cond-mat.str-elCross-listedarXiv:2603.05707•Physical Review Research•Mar 9, 2026•0 citations

Distinguishing phases via non-Markovian dynamics of entanglement in topological quantum codes under parallel magnetic field

K J Harikrishnan, Amit Kumar Pal

We investigate the static and the dynamical behavior of localizable entanglement and its lower bounds on nontrivial loops of topological quantum codes with parallel magnetic field. Exploiting the connection between the stabilizer states and graph states in the absence of the parallel field and external noise, we identify a specific measurement basis, referred to as the canonical measurement basis, that optimizes localizable entanglement when measurement is restricted to single-qubit Pauli measurements only, thereby providing a lower bound. In situations where computing even the lower bound is difficult, we propose an approximation of the lower bound that can be computed for larger systems according to the computational resource in hand. Additionally, we compute a lower bound of the localizable entanglement that can be computed by determining the expectation value of an appropriately designed witness operator. We investigate the behavior of these lower bounds in the vicinity of the topological to nontopological quantum phase transition of the system, and perform a finite-size scaling analysis. We also investigate the dynamical features of these lower bounds when the system is subjected to Markovian or non-Markovian single-qubit dephasing noise. We find that in the case of the non-Markovian dephasing noise, at large time, the canonical measurement-based lower bound oscillates with a larger amplitude when the initial state of the system undergoing dephasing dynamics is chosen from the nontopological phase, compared to the same for an initial state from the topological phase. On the other hand, repetitive collapses followed by revivals to high value with time are observed for the proposed witness-based lower bound in the nontopological phase, which is absent in the topological phase. These features can be utilized to distinguish the topological phase of the system from the nontopological phase in the presence of dephasing noise.

arXiv:2108.11198•Physical Review A•May 16, 2022•5 citations

Universal fidelity near quantum and topological phase transitions in finite one-dimensional systems

Elio J. König, Alex Levchenko, Nicholas Sedlmayr

We study the quantum fidelity (ground-state overlap) near quantum phase transitions of the Ising universality class in one-dimensional (1D) systems of finite-size $L$. Prominent examples occur in magnetic systems (e.g., spin-Peierls, the anisotropic $XY$ model) and in 1D topological insulators of any topologically nontrivial Altland-Zirnbauer-Kitaev universality class. The rescaled fidelity susceptibility is a function of the only dimensionless parameter $LM$, where $2M$ is the gap in the fermionic spectrum. We present analytic expressions for the fidelity susceptibility for periodic and open boundaries conditions with zero, one, or two edge states. The latter are shown to have a crucial impact and alter the susceptibility both quantitatively and qualitatively. We support our analytical solutions with numerical data.

arXiv:1602.04201•Physical review. B./Physical review. B•Jun 29, 2016•22 citations

Quantum phase transition as an interplay of Kitaev and Ising interactions

A. Langari, Amir Mohammadaghaei, Reza Haghshenas

We study the interplay between the Kitaev and Ising interactions on a ladder geometry. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. The nature of the SPT phase is confirmed by degeneracy of the entanglement spectrum. Nonlocal order parameters that indirectly measure phase factors (inequivalent projective representations of the symmetries) of the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry explicitly show the protection of the SPT phase under the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. We derive the effective theory to describe the topological phase transition on the ladder geometry, which is given by a transverse field Ising model with/without next-nearest-neighbor coupling based on the primary Ising configurations. The ladder has three phases, namely, the Kitaev SPT, symmetry-broken ferro/antiferromagnetic order, and classical spin liquid. The nonzero quantum critical point and its corresponding central charge are provided by the effective theory, which are in full agreement with the numerical results, i.e., the divergence of entanglement entropy at the critical point and change of the entanglement spectrum degeneracy. The central charge of the critical points are either $c=1$ or $c=2$, with the magnetization and correlation exponents being 1/4 and 1/2, respectively. The transition from the classical spin-liquid phase of the frustrated Ising ladder to the Kitaev SPT phase is mediated by a floating phase, which shows strong finite entanglement scaling.

arXiv:1410.2025•Physical Review B•Jan 15, 2015•8 citations

Multipartite entanglement of the topologically ordered state in a perturbed toric code

Yu-Ran Zhang, Yu Zeng, Tao Liu +3 more

We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information, can characterize topological quantum phase transitions in the spin-$\frac{1}{2}$ toric code model (TCM) on a square lattice with external fields. We show that the quantum Fisher information density of the ground state can be written in terms of the expectation values of gauge-invariant Wilson loops for different sizes of square regions and identify ${\mathbb{Z}}_{2}$ topological order by its scaling behavior. Furthermore, we use this multipartite entanglement witness to investigate thermalization and disorder-assisted stabilization of topological order after a quantum quench. Moreover, with an upper bound of the quantum Fisher information, we demonstrate the absence of finite-temperature topological order in the two-dimensional TCM in the thermodynamic limit. Our results provide insights to topological phases, which are robust against external disturbances and are candidates for topologically protected quantum computation.

arXiv:2109.03315•Physical Review Research•May 24, 2022•13 citations
We investigate the topological-to-nontopological quantum phase transitions (QPTs) occurring in the Kitaev code under local perturbations in the form of a local magnetic field and spin-spin interactions of the Ising-type using fidelity susceptibility (FS) and entanglement as the probes. We assume the code to be embedded on the surface of a wide cylinder of height MMM and circumference DDD with M\ensuremath≪DM\ensuremath{\ll}DM\ensuremath≪D. We demonstrate a power-law divergence of FS across the QPT and determine the quantum critical points via a finite-size-scaling analysis. We verify these results by mapping the perturbed Kitaev code to the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions and computing the single-site magnetization as the order parameter using the quantum Monte Carlo technique. We also demonstrate a finite-size odd-even dichotomy in the occurrence of the QPT in the Kitaev ladder with respect to the odd and even values of DDD, when the system is perturbed with only Ising interaction. Our results also indicate a higher robustness of the topological phase of the Kitaev code against local perturbations if the boundary is made open along one direction. We further consider a local entanglement witness operator designed specifically to capture a lower bound to the localizable entanglement on the vertical nontrivial loop of the code. We show that the first derivative of the expectation value of the witness operator exhibits a logarithmic divergence across the QPT and perform the finite-size-scaling analysis. We demonstrate similar behavior of the expectation value of the appropriately constructed witness operator also in the case of a locally perturbed color code with open boundaries.
The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.
We investigate the impact of an isotropic antiferromagnetic Heisenberg perturbation on the toric code, focusing on the resulting quantum phase transition and the nature of the phase that emerges beyond topological order. Using neural-network quantum states (NQS), we compute ground states over a wide range of Heisenberg couplings while fully respecting the exact symmetries of the model. In the weak-coupling regime, the numerical results are in excellent agreement with an effective low-energy description derived from a Schrieffer-Wolff (SW) transformation, providing analytic control over the perturbative breakdown of topological order. We show that the Heisenberg perturbation only renormalizes local operators at low orders, whereas mixing between topological sectors occurs only at a perturbative order proportional to the system size. At intermediate values of the Heisenberg interaction, the topological phase breaks down. We estimate the critical point through a combination of the fidelity susceptibility and the logarithmic susceptibility of non-contractible Wilson loops for various system sizes. Furthermore, we utilize the topological entanglement entropy to provide a comprehensive characterization of the phase transition. Beyond the transition, an antiferromagnetic ±X/±Z\pm X/\pm Z±X/±Z Néel phase emerges, characterized by a fourfold-degenerate symmetry-broken manifold, which is explicitly probed using staggered-magnetization-based diagnostics. Our results show how local two-spin interactions, which naturally arise in realistic implementations of the toric code, drive the breakdown of topological order. Moreover, we establish the SW approach as a systematic framework for analyzing such perturbations in combination with variational many-body methods.
We investigate the static and the dynamical behavior of localizable entanglement and its lower bounds on nontrivial loops of topological quantum codes with parallel magnetic field. Exploiting the connection between the stabilizer states and graph states in the absence of the parallel field and external noise, we identify a specific measurement basis, referred to as the canonical measurement basis, that optimizes localizable entanglement when measurement is restricted to single-qubit Pauli measurements only, thereby providing a lower bound. In situations where computing even the lower bound is difficult, we propose an approximation of the lower bound that can be computed for larger systems according to the computational resource in hand. Additionally, we compute a lower bound of the localizable entanglement that can be computed by determining the expectation value of an appropriately designed witness operator. We investigate the behavior of these lower bounds in the vicinity of the topological to nontopological quantum phase transition of the system, and perform a finite-size scaling analysis. We also investigate the dynamical features of these lower bounds when the system is subjected to Markovian or non-Markovian single-qubit dephasing noise. We find that in the case of the non-Markovian dephasing noise, at large time, the canonical measurement-based lower bound oscillates with a larger amplitude when the initial state of the system undergoing dephasing dynamics is chosen from the nontopological phase, compared to the same for an initial state from the topological phase. On the other hand, repetitive collapses followed by revivals to high value with time are observed for the proposed witness-based lower bound in the nontopological phase, which is absent in the topological phase. These features can be utilized to distinguish the topological phase of the system from the nontopological phase in the presence of dephasing noise.
We study the quantum fidelity (ground-state overlap) near quantum phase transitions of the Ising universality class in one-dimensional (1D) systems of finite-size LLL. Prominent examples occur in magnetic systems (e.g., spin-Peierls, the anisotropic XYXYXY model) and in 1D topological insulators of any topologically nontrivial Altland-Zirnbauer-Kitaev universality class. The rescaled fidelity susceptibility is a function of the only dimensionless parameter LMLMLM, where 2M2M2M is the gap in the fermionic spectrum. We present analytic expressions for the fidelity susceptibility for periodic and open boundaries conditions with zero, one, or two edge states. The latter are shown to have a crucial impact and alter the susceptibility both quantitatively and qualitatively. We support our analytical solutions with numerical data.
We study the interplay between the Kitaev and Ising interactions on a ladder geometry. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. The nature of the SPT phase is confirmed by degeneracy of the entanglement spectrum. Nonlocal order parameters that indirectly measure phase factors (inequivalent projective representations of the symmetries) of the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry explicitly show the protection of the SPT phase under the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. We derive the effective theory to describe the topological phase transition on the ladder geometry, which is given by a transverse field Ising model with/without next-nearest-neighbor coupling based on the primary Ising configurations. The ladder has three phases, namely, the Kitaev SPT, symmetry-broken ferro/antiferromagnetic order, and classical spin liquid. The nonzero quantum critical point and its corresponding central charge are provided by the effective theory, which are in full agreement with the numerical results, i.e., the divergence of entanglement entropy at the critical point and change of the entanglement spectrum degeneracy. The central charge of the critical points are either c=1c=1c=1 or c=2c=2c=2, with the magnetization and correlation exponents being 1/4 and 1/2, respectively. The transition from the classical spin-liquid phase of the frustrated Ising ladder to the Kitaev SPT phase is mediated by a floating phase, which shows strong finite entanglement scaling.
We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information, can characterize topological quantum phase transitions in the spin-12\frac{1}{2}21​ toric code model (TCM) on a square lattice with external fields. We show that the quantum Fisher information density of the ground state can be written in terms of the expectation values of gauge-invariant Wilson loops for different sizes of square regions and identify Z2{\mathbb{Z}}_{2}Z2​ topological order by its scaling behavior. Furthermore, we use this multipartite entanglement witness to investigate thermalization and disorder-assisted stabilization of topological order after a quantum quench. Moreover, with an upper bound of the quantum Fisher information, we demonstrate the absence of finite-temperature topological order in the two-dimensional TCM in the thermodynamic limit. Our results provide insights to topological phases, which are robust against external disturbances and are candidates for topologically protected quantum computation.