Generalized toric codes coupled to thermal baths

Abstract

We have studied the dynamics of a generalized toric code based on qudits at finite temperature by finding the master equation coupling the code's degrees of freedom to a thermal bath. We find that in the case of qutrits, new types of anyons and thermal processes appear that are forbidden for qubits. These include creation, annihilation and diffusion throughout the system code. It is possible to solve the master equation in a short-time regime and find expressions for the decay rates as a function of the dimension d of the qudits. While we provide an explicit proof that the system relaxes to the Gibbs state for arbitrary qudits, we also prove that above a certain crossover temperature the qutrits' initial decay rate is smaller than the original case for qubits. Surprisingly, this behavior only happens for qutrits and not for other qudits with d > 3.3 the property of locality in error detection and correction is of great importance both theoretically and for practical implementations [12][13][14]. It is also possible to generalize these topological codes for units of quantum information based on multilevel systems known as qudits, i.e. d-level systems [15][16][17][18], and study their local stability [19]. An alternative scheme to manipulate topological quantum information is based not in the ground-state properties of the system but in its excitations [11]. These are non-Abelian anyons that can implement universal gates for quantum information [20]. However, being within the framework of topological codes based on ground state properties, it is possible to formulate new surface codes known as topological color codes (TCCs) [21] such that they have enhanced quantum computational capabilities while preserving nice locality properties [21][22][23]. TCCs in two-dimensional (2D) surfaces allow for the implementation of quantum gates in the whole Clifford group. This makes possible quantum teleportation, distillation of entanglement and dense coding in a fully topological scenario. Moreover, with TCCs in 3D spatial manifolds it is possible to implement the quantum gate π/8, thereby allowing for universal quantum computation [24,25]. Very nice applications of topological surface codes can be seen in other fields [26,27].Acting externally on topological codes, in order to cure the system from external noise and decoherence, produces benefits from the locality properties of these codes. Namely, a very important figure of merit is the error threshold of the topological code, i.e. the critical value of the external noise below which it is possible to perform quantum operations with arbitrary accuracy and time. For toric codes with qubits, the error threshold is very good, about 11% [12]. This value is obtained by mapping the process of error correction to a classical Ising model on a 2D lattice with random bonds. Interestingly enough, this type of mapping can be made more general and applied to TCCs yielding the same error threshold [28] while maintaining enhanced quantum capabilities [29,30]. These results have been confirm...

Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime

“…We have identified a low temperature regime in which the relaxation dynamics are dominated by topologically non-trivial random walks of quasiparticle pairs; consequently the finite-size and temperature scaling of this regime are distinct from the high temperature regime above the crossover temperature. We find that both the finite-size and finite temperature scaling are stronger in this low temperature regime than at higher temperatures where the behavior coincides with the expected scaling in the thermodynamic limit 45 .…”

Quantum memories at finite temperature

“…A further physically motivated study includes the rigorous proof of instability in the toric code model using the Liouvillian open dynamics to show that the expectation values of the logical operators of the toric code model decay rapidly when weakly coupled to a Markovian environment ). These results are generalized by Chesi et al (2010) and are considered for the toric code with higher-dimensional spins in Viyuela, Rivas, and Martin-Delagado (2012).…”

“…An extensive program of research has shown that we can model thermal dynamics of a many-body quantum memory with a simple rate equation (Davies, 1974;Horodecki, 2007, 2009;Alicki et al, 2010;Chesi, Röthlisberger, and Loss, 2010;Alicki, 2012;Viyuela, Rivas, and Martin-Delagado, 2012;Weiss, 2012). These methods are built from the discovery of exact master equations to study dissipation, a study initially pioneered by Caldeira and Leggett (Calderia and Leggett, 1981;Leggett et al, 1987;DiVincenzo and Loss, 2005).…”

Necessity of an energy barrier for self-correction of Abelian quantum doubles

“…(21), one is left with a Lindbladian that can be written as the sum of local terms as done, e.g., in Ref. [22]. Note, however, that we are taking another approach, as the representation mentioned above is not particularly helpful for our derivation of the spectral gap as it obfuscates the underlying general algebraic structure.…”