Candidate for a passively protected quantum memory in two dimensions

An interesting problem in the field of quantum error correction involves finding a physical system that hosts a ``passively-protected quantum memory,'' defined as an encoded qubit coupled to an environment that naturally wants to correct errors. To date, a quantum memory stable against finite-temperature effects is only known in four spatial dimensions or higher. Here, we take a different approach to realize a stable quantum memory by relying on a driven-dissipative environment. We propose a new model which appears to passively correct against both bit-flip and phase-flip errors in two dimensions: A square lattice composed of photonic ``cat qubits'' coupled via dissipative terms which tend to fix errors locally. Inspired by the presence of two distinct $mathbb{Z}_2$-symmetry-broken phases, our scheme relies on Ising-like dissipators to protect against bit flips and on a driven-dissipative photonic environment to protect against phase flips. We also connect the ability to store the quantum memory to the existence of a non-equilibrium phase in the photonic-Ising model, hinting a perturbative stability under more general noise channels. At the end, we discuss possible ways to realize the photonic-Ising model.