Yijia Xu

We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The n-rotor Clifford group, U(1)n(n+1)/2 ⋊ GLn(Z), is represented by continuous U(1) gates generated by polynomials quadratic in angular momenta, as well as discrete GLn(Z) gates generated by momentum sign-flip and sum gates. Understandings in rotor Clifford group allow us to establish connections between homological rotor error-correcting codes and oscillator quantum codes, including Gottesman-Kitaev-Preskill codes and rotation-symmetric bosonic codes. Inspired by homological rotor codes, we provide a systematic construction of multi-mode rotation-symmetric bosonic codes by analoging Fock states to rotor states with non-negative angular momentum. This new family of multi-mode bosonic codes protect against dephasing and changes in occupation numbers, which we call homological number-phase codes. Their encoding and decoding circuits are readily derived from the corresponding rotor Clifford operations. 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. References: arXiv:2311.07679, homological rotor error-correcting codes (arXiv:2303.13723), GKP-stabilizer codes (arXiv:1903.12615)