The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection ("self-correction"), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose two-dimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive long-range interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive long-range nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that self-correcting quantum memories can exist in 2D at finite temperatures.

We numerically study the effects of two forms of quenched disorder on the anyons of the toric code. Firstly, a new class of codes based on random lattices of stabilizer operators is presented, and shown to be superior to the standard square lattice toric code for certain forms of biased noise. It is further argued that these codes are close to optimal, in that they tightly reach the upper bound of error thresholds beyond which no correctable CSS codes can exist. Additionally, we study the classical motion of anyons in toric codes with randomly distributed onsite potentials. In the presence of repulsive long-range interaction between the anyons, a surprising increase with disorder strength of the lifetime of encoded states is reported and explained by an entirely incoherent mechanism. Finally, the coherent transport of the anyons in the presence of both forms of disorder is investigated, and a significant suppression of the anyon motion is found.

We investigate the creation of highly entangled ground states in a system of three exchange-coupled qubits arranged in a ring geometry. Suitable magnetic field configurations yielding approximate Greenberger-Horne-Zeilinger and exact W ground states are identified. The entanglement in the system is studied at finite temperature in terms of the mixed-state tangle tau. By generalizing a conjugate gradient optimization algorithm originally developed to evaluate the entanglement of formation, we demonstrate that tau can be calculated efficiently and with high precision. We identify the parameter regime for which the equilibrium entanglement of the tripartite system reaches its maximum.

We present two ready-to-use numerical algorithms to evaluate convex-roof extensions of arbitrary pure-state entanglement monotones. Their implementation leaves the user merely with the task of calculating derivatives of the respective pure-state measure. We provide numerical tests of the algorithms and demonstrate their good convergence properties. We further employ them in order to investigate the entanglement in particular few-spins systems at finite temperature. Namely, we consider ferromagnetic Heisenberg exchange-coupled spin-1 2 rings subject to an inhomogeneous inplane field geometry obeying full rotational symmetry around the axis perpendicular to the ring through its center. We demonstrate that highly entangled states can be obtained in these systems at sufficiently low temperatures and by tuning the strength of a magnetic field configuration to an optimal value which is identified numerically.

We study the two-dimensional toric code Hamiltonian with effective long-range interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow to increase the lifetime of the stored quantum information arbitrarily by making $L$, the linear size of the memory, larger [Phys. Rev. A 82 022305 (2010)]. We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough $L$. This influence can be both beneficial or detrimental. In particular, we study an effective Hamiltonian proposed in [Phys. Rev. B 83 115415 (2011)] that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in $L^2$ for both periodic and open boundary conditions, though the exponent in the latter case is found to be less favourable. However, $L$ is upper-bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.Comment: 14+2 pages, 10 figures, revtex; v2: author added; v3: minor improvements, to appear in Phys. Rev.