Performance analysis on generalized dark spin-cat encoding in atoms

Different quantum platforms often exhibit distinct noise mechanisms, motivating the tailored encoding strategies at the physical level to shape errors into forms that are easier to correct. We have developed a spin-cat encoding scheme within the ground state Zeeman levels in neutral atoms [arXiv: 2408.04421]. In this scheme, the code subspace is spanned by two spin-coherent states (SCS) located at antipodal points on the generalized Bloch sphere (GBS), and it is dissipatively stabilized through coupling to excited levels. The bit-flip rate for this encoding scheme decreases exponentially with the size of the Zeeman manifold, while the dephasing error only grows polynomially.



Notably, the two SCS can in general be stabilized at arbitrary points on the GBS, enabling control over the noise bias. We develop a semiclassical method to accurately estimate the bit-flip rate in this general setting and explore strategies to modify the code states that reduce the tunnelling between them. Furthermore, we discuss the possibility of stabilizing the new code states by engineering colored dissipation. This generalized framework will enhance the controllability under spin-cat encoding and provide flexibility in choosing the desired error bias in practice.