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arXiv:2102.03046

Protecting topological order by dynamical localization

Yu Zeng, Alioscia Hamma, Yu-Ran Zhang, Junpeng Cao, Heng Fan, Wu-Ming Liu

Published Feb 5, 2021• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2102.03046

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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.
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 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.
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.
Topological phases of matter are considered the bedrock of novel quantum materials as well as ideal candidates for quantum computers that possess robustness at the physical level. The robustness of the topological phase at finite temperature or away from equilibrium is therefore a very desirable feature. Disorder can improve the lifetime of the encoded topological qubits. Here we tackle the problem of the survival of the topological phase as detected by topological entropy, after a sudden quantum quench. We introduce a method to study analytically the time evolution of the system after a quantum quench and show that disorder in the couplings of the Hamiltonian of the toric code and the resulting Anderson localization can make the topological entropy resilient.
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.
The ground-state degeneracy of topologically ordered gapped Hamiltonians is the bedrock for self-correcting quantum memories, which are unfortunately not stable away from equilibrium even at zero temperature. This plague precludes practical robust self-correction since stability at zero temperature is a prerequisite for finite-temperature robustness. In this work, we show that the emergence of a bounded light cone renders the unitary time evolution a quasiadiabatic continuation that preserves topological order, with the initial ground space retaining its macroscopic distance at all times as a quantum code. We also show how bounded light cones can emerge through suitable perturbations in Kitaev's toric code and honeycomb model. Our results suggest that topological orders and self-correcting quantum memories can be dynamically robust at zero temperature.