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arXiv:1603.01546 · Physical Review A

Thermal stability of the two-dimensional topological color code

Razieh Mohseninia

Published Aug 8, 2016• 1 citation
arXivHTMLPDF

Abstract

Thermal stability of the topological color code in the presence of a thermal bath is studied. We study the Lindblad evolution of the observables in the weak-coupling limit of the Born-Markov approximation. The autocorrelation functions of the observables are used as a figure of merit for the thermal stability. We show that all of the observables autocorrelation functions decay exponentially in time. By finding a lower bound of the decay rate, which is a constant independent of the system size, we show that the topological color code is unstable against thermal fluctuations from the bath at finite temperature, even though it is stable at $T=0$ against local quantum perturbations.

DOI
10.1103/physreva.94.022306

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Thermal stability of the topological color code in the presence of a thermal bath is studied. We study the Lindblad evolution of the observables in the weak-coupling limit of the Born-Markov approximation. The autocorrelation functions of the observables are used as a figure of merit for the thermal stability. We show that all of the observables autocorrelation functions decay exponentially in time. By finding a lower bound of the decay rate, which is a constant independent of the system size, we show that the topological color code is unstable against thermal fluctuations from the bath at finite temperature, even though it is stable at T=0T=0T=0 against local quantum perturbations.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.
We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H0, we prove that there exists a constant threshold ϵ>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.
We analyse stability of the four-dimensional Kitaev model — a candidate for scalable quantum memory — in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of qubit for statistical mechanics.
The color code model is a crucial instance of a Calderbank-Shor-Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local C∗C^{*}C∗-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by Rep(D(Z2×Z2))≃Rep(D(Z2))⊠Rep(D(Z2))\mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2)) \simeq \mathsf{Rep}(D(\mathbb{Z}_2)) \boxtimes \mathsf{Rep}(D(\mathbb{Z}_2))Rep(D(Z2​×Z2​))≃Rep(D(Z2​))⊠Rep(D(Z2​)), which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices. We also establish the existence and uniqueness of the KMS state at every finite inverse temperature and derive an explicit description of the corresponding thermal distribution of stabilizer excitations.