Covariant Quantum Error-Correcting Codes with Metrological Entanglement Advantage

We show that a subset of the basis for the irreducible representations of the total SU(2) rotation forms a covariant approximate quantum error-correcting code with transversal U(1) logical gates. Using only properties of the angular momentum algebra, we obtain bounds on the code inaccuracy against generic noise on any known d sites and against heralded d-local erasures, generalizing and improving previous works on the ``thermodynamic code" to general local spin and different irreducible representations. We demonstrate that this family of codes can host and protect a probe state with quantum Fisher information surpassing the standard quantum limit when the sensing parameter couples to the generator of the U(1) logical gate.