Self correction requires Energy Barrier for Abelian quantum doubles

Abstract: We rigorously establish an Arrhenius law for the mixing time of quantum doubles based on any Abelian group $\mathbb{Z}_d$. We have made the concept of the energy barrier therein mathematically well-defined, it is related to the minimum energy cost the environment has to provide to the system in order to produce a generalized Pauli error, maximized for any generalized Pauli errors, not only logical operators. We evaluate this generalized energy barrier in Abelian quantum double models and find it to be a constant independent of system size. Thus, we rule out the possibility of entropic protection for this broad group of models.

• The paper establishes that achieving self-correction in Abelian quantum double models fundamentally requires the presence of a nontrivial energy barrier.
• A key finding is that for Abelian quantum double models, this required energy barrier is a constant independent of system size, limiting the system's thermal memory time.
• This research challenges previous suggestions of entropic protection for self-correction in these models and highlights the need to explore non-Abelian or alternative constructions.

Energy Barriers and Self-Correction in Abelian Quantum Double Models

The paper "Self correction requires Energy Barrier for Abelian quantum doubles" explores a fundamental question in quantum computing: the preservation of arbitrary quantum information in a thermal environment without active error correction. It investigates whether quantum double models based on Abelian groups can achieve self-correction, thereby maintaining encoded quantum information over extended periods. The authors rigorously establish that any such self-correcting property necessitates a nontrivial energy barrier.

Theoretical Framework and Main Results

The authors employ the Abelian quantum double models, specifically those based on the cyclic group $\mathbb{Z}_d$, a generalization of Kitaev's toric code. The research pivots on characterizing and rigorously defining the energy barrier for these models, tying it directly to the Arrhenius law, which dictates that the memory time is related exponentially to the energy barrier and the inverse temperature.

A key outcome of the study is the finding that for any Abelian quantum double model, the generalized energy barrier is constant and independent of system size. This implies that the system’s memory time scales exponentially with inverse temperature but cannot exceed an upper bound determined by this constant energy barrier. The absence of a scaling energy barrier precludes the possibility of entropic protection in these models.

Numerical Uniformity and Thermodynamic Implications

The research presented in this paper challenges claims made by previous studies, such as the entropic protection scenario proposed by Brown et al., where it was suggested that system configurations could lead to super-exponential growth of memory time with inverse temperature. The models of Brown et al., based on the quantum double of $\mathbb{Z}_5$ enhanced with defect lines, were previously argued to leverage entropic characteristics. However, the current results negate this possibility for any two-dimensional Abelian quantum double construction, providing a concrete bound for the system's thermal stability.

Future Directions

The findings of this paper open multiple avenues for future research. Firstly, it prompts exploration into non-Abelian quantum double models which are not presently constrained by the same bounds on energy barriers. These models might hold potential for discovering truly self-corrective quantum memories. Additionally, investigating alternative noise models or environmental interactions could reveal new dynamics where entropic protection might play a role.

This work is instrumental in clarifying the structural requirements for achieving quantum self-correction and suggests that any hope for such systems must critically reassess the interplay between energy barriers, system dimensionality, and topological properties beyond Abelian constructions. The authors also suggest investigating mechanisms that could potentially lower the dimensionality requirements while still retaining necessary energy barriers.