Multimode rotation-symmetric bosonic codes from homological rotor codes

We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The $n$-rotor Clifford group, $\text{U}{\left(1\right)}^{n(n+1)/2}⋊{\text{GL}}_n\left(\mathbb{Z}\right)$, is represented by continuous $\text{U}\left(1\right)$ gates generated by polynomials quadratic in angular momenta, as well as discrete ${\text{GL}}_n\left(\mathbb{Z}\right)$ gates generated by momentum sign-flip and sum gates. Our understanding of this group allows us to establish connections between homological rotor error-correcting codes [Vuillot, Ciani, and Terhal, Commun. Math. Phys. 405, 53 (2024)] 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 multimode rotation-symmetric bosonic codes by making a parallel between oscillator Fock states and rotor states with fixed non-negative angular momentum. This family of homological number-phase codes protects against dephasing and changes in occupation number. Encoding and decoding circuits for these codes can be derived from the corresponding rotor Clifford operations. As a result of independent interest, we show how to nondestructively measure the oscillator phase using conditional occupation-number addition and postselection. We also outline several rotor and oscillator varieties of the Gottesman-Kitaev-Preskill-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020).].