Covariant Quantum Error-Correcting Codes with Metrological Entanglement Advantage

Abstract: 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.