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arXiv:2109.03315 · Physical Review Research

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

Yu-Ran Zhang, Yu Zeng, Tao Liu, Heng Fan, J. Q. You, Franco Nori

Published May 24, 2022• 13 citations
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Abstract

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.

DOI
10.1103/physrevresearch.4.023144

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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.
We demonstrate that multipartite entanglement is able to characterize one-dimensional topological order, which is witnessed by the scaling behavior of the quantum Fisher information of the ground state with respect to the spin operators defined in the original and dual lattices. We investigate an exactly-solvable long-range extended spin-12\frac{1}{2}21​ Ising model with nontrivial Z\mathbf{Z}Z-symmetry topological order identified equivalently by winding numbers and paired Majorana zero modes at each end. For topologically ordered phases with high winding numbers, we can define a Z\mathbf{Z}Z topological invariant by the scaling coefficient of the dual quantum Fisher information density. Containing richer properties and more complex structures than bipartite entanglement, the dual multipartite entanglement of the topologically ordered state has promising applications in robust quantum computation and quantum metrology, and could be generalized to higher-dimensional systems.
We demonstrate that multipartite entanglement is able to characterize one-dimensional symmetry-protected topological order, which is witnessed by the scaling behavior of the quantum Fisher information of the ground state with respect to the spin operators defined in the dual lattice. We investigate an extended Kitaev chain with a Z symmetry identified equivalently by winding numbers and paired Majorana zero modes at each end. The topological phases with high winding numbers are detected by the scaling coefficient of the quantum Fisher information density with respect to generators in different dual lattices. Containing richer properties and more complex structures than bipartite entanglement, the dual multipartite entanglement of the topological state has promising applications in robust quantum computation and quantum metrology, and can be generalized to identify topological order in the Kitaev honeycomb model.
The topological order of the Kitaev toric code is insensitive to disorder in its couplings since all star and plaquette operators commute; consequently, the stabilizer ground-state manifold is independent of the individual coupling strengths, and the phase remains stable against weak local perturbations. This insensitivity, however, is a property of the eigenstates and not of the dynamics they generate, and we show that the disorder sensitivity of a probe prepared outside that eigenbasis can be a metrological resource. In particular, we investigate the estimation of the strength of three topological-order-breaking perturbations, the nonlinear Castelnovo-Chamon deformation, nearest-neighbor Ising interactions, and a magnetic field along horizontal edges, encoded on an initially separable probe through unitary evolution under the disordered toric code Hamiltonian. We demonstrate that the quenched average quantum Fisher information (QFI), a figure of merit for metrological precision, exceeds that of its ordered counterpart in suitable regimes of perturbation strength and encoding time. For the ordered case, we further identify optimal product probes, for which the quadratic and cubic terms of the short-time expansion vanish identically, leaving a transient quartic scaling of QFI. Interestingly, this window persists as long as the multipartite entanglement generated by the encoding continues to increase, and it survives under local dephasing, bit-flip, and amplitude damping noise acting independently on each site, with the non-unital channel being the least detrimental.
We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly coupled ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ and the strongly coupled Z2{\mathbb{Z}}_{2}Z2​ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional Ising∗{\mathrm{Ising}}^{*}Ising∗ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local Z2{\mathbb{Z}}_{2}Z2​ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.
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.
As a prototype model of topological quantum memory, two-dimensional toric code is genuinely immune to generic local static perturbations, but fragile at finite temperature and also after non-equilibrium time evolution at zero temperature. We show that dynamical localization induced by disorder makes the time evolution a local unitary transformation at all times, which keeps topological order robust after a quantum quench. We verify this conclusion by investigating the Wilson loop expectation value and topological entanglement entropy. Our results suggest that the two dimensional topological quantum memory can be dynamically robust at zero temperature.