On Thermal Stability of Topological Qubit in Kitaev's 4D Model

Abstract

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.

Quantum memories at finite temperature

“…They found that, while some topological order parameters remain robust up to a critical temperature, a qubit cannot be stored in the three-dimensional toric code at finite temperature because looplike order parameters decay rapidly in the large system-size limit. Similar results are anticipated using the methods of Alicki et al (2010), where they studied the thermal dynamics of the three-dimensional toric code model by considering the model weakly coupled to a Markovian thermal reservoir.…”

“…Indeed, the assumptions necessary to prove the discovered threedimensional no-go theorems describe a limited set of models when compared with the theorems known in two dimensions. Supporting these results, we also have various physical results showing that topological entanglement entropy (Castelnovo and Chamon, 2008), and the correlation functions of stringlike logical operators (Alicki et al, 2010) decay rapidly for the three-dimensional toric code model. We see in Sec.…”

“…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).…”

“…While unphysical due to its nonlocal Hamiltonian interactions, the Curie-Weiss model is a simple classical model that enables a detailed analysis of the contribution of both the energy barrier and the free energy that can be used to determine its coherence time (Alicki and Horodecki, 2006). For a detailed discussion on the Curie-Weiss model see Kochmański, Paszkiewicz, and Wolski (2013) and references therein.…”

“…To this end, in the thermodynamic limit, and at suitably low temperatures, the Curie-Weiss model is able to robustly encode a classical bit of information for arbitrarily long time scales. This is shown in Alicki and Horodecki (2006) by taking x to the continuum limit and applying Kramer's formula (Gardiner, 1983).…”