Clifford operations and homological codes for rotors and oscillators

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