Abstract: A set of quantum error correcting codes for spin systems providing a path to fault tolerance with fewer qubits by using the extra available internal degrees of freedom. Cat codes in the spin system are used which are the equal superpositions of stretched states, with consideration of errors that are products of angular momentum and not just restricted to linear in angular momentum operators.

Abstract: A method for preparing an entangled quantum state of an atomic ensemble is provided. The method includes loading each atom of the atomic ensemble into a respective optical trap; placing each atom of the atomic ensemble into a same first atomic quantum state by impingement of pump radiation; approaching the atoms of the atomic ensemble to within a dipole-dipole interaction length of each other; Rydberg-dressing the atomic ensemble; during the Rydberg-dressing operation, exciting the atomic ensemble with a Raman pulse tuned to stimulate a ground-state hyperfine transition from the first atomic quantum state to a second atomic quantum state; and separating the atoms of the atomic ensemble by more than a dipole-dipole interaction length.