Designing Codes around Interactions: The Case of a Spin

I present a new approach for designing quantum error-correcting codes guaranteeing a physically natural implementation of Clifford operations. Inspired by the scheme put forward by Gottesman, Kitaev, and Preskill for encoding a qubit in an oscillator in which Clifford operations may be performed via Gaussian unitaries, this approach yields new schemes for encoding a qubit in a large spin in which single-qubit Clifford operations may be performed via spatial rotations. I construct all possible examples of such codes, provide universal-gate-set implementations using quadratic angular-momentum Hamiltonians, and derive criteria for when these codes exactly correct physically relevant errors.