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We develop quantum information processing primitives for the planar rotor,the state space of a particle on a circle. By interpreting rotor wavefunctionsas periodically identified wavefunctions of a harmonic oscillator, we determinethe group of bosonic Gaussian operations inherited by the rotor. This $n$-rotorClifford group, $\text{U}(1)^{n(n+1)/2} \rtimes \text{GL}_n(\mathbb{Z})$, isrepresented by continuous $\text{U}(1)$ gates generated by polynomialsquadratic in angular momenta, as well as discrete $\text{GL}_n(\mathbb Z)$momentum sign-flip and sum gates. We classify homological rotorerror-correcting codes [arXiv:2303.13723] and various rotor states based onequivalence under Clifford operations.  Reversing direction, we map homological rotor codes and rotor Cliffordoperations back into oscillators by interpreting occupation-number states asrotor states of non-negative angular momentum. This yields new multimodehomological bosonic codes protecting against dephasing and changes inoccupation number, along with their corresponding encoding and decodingcircuits. In particular, we show how to non-destructively measure theoscillator phase using conditional occupation-number addition and postselection. We also outline several rotor and oscillator varieties of theGKP-stabilizer codes [arXiv:1903.12615].

Clifford operations and homological codes for rotors and oscillators