Abstract

We develop quantum information processing primitives for the planar rotor, the state space of aparticle on a circle. By interpreting rotor wavefunctions as periodically identified wavefunctions of aharmonic oscilator, we determine the group of bosonic Gaussian operations inherited by the rotor.This n-rotor Cliford group, U(1)n(n+1)/2xGLn(Z),is represented by continuous U(1) gatesgenerated by polynomials quadratic in angular momenta, as well as discrete GLn(Z) momentumsign-flip and sum gates. We classify homological rotor error-correcting codes [arXiv:2303.13723]and various rotor states based on eguivalence under Cliford operations. Reversing direction, wemap homological rotor codes and rotor Clifford operations back into oscillators by interpretingoccupation-number states as rotor states of non-negative angular momentum. This yields newmultimode homological bosonic codes protecting aaainst dephasing and changes in occupationnumber, along with their corresponding encoding and decoding circuits. in particular, we show howto non-destructively measure the oscillator phase using conditional occupation-number addition andpost selection. We also outline several rotor and oscillator varieties of the GKP-stabilizer codes[arXiv:1903.126151.