Optimal encoding of two dissipative interacting qubits

We investigate a system of two coupled qubits interacting with an Ohmic bath as a physical model for the implementation of one logical qubit. In this model, the interaction with the other qubit represents unitary noise, while the Ohmic bath is responsible for finite temperature. In the presence of a one-dimensional decoherence-free subspace (DFS), we show that, while this is not sufficient to protect a qubit from decoherence, it can be exploited to encode one logical qubit with greater performance than the physical one. We show different possible strategies for the optimal encoding of a logical qubit through a numerical analysis based on matrix product states. This method reproduces faithfully the results of perturbative calculations, but it can be extended to cases of crucial interest for physical implementations, e.g., in the case of strong coupling with the bath. As a result, a logical qubit encoded in the subspace which is the direct sum of the antiferromagnetic states in the Bell basis, namely the DFS and the antiferromagnetic state in the triplet, is the optimally robust one, as it takes advantage of both the anchoring to the DFS and the protection from the antiferromagnetic interaction.