We formally define homological quantum rotor codes which use multiple quantum
rotors to encode logical information. These codes generalize homological or CSS
quantum codes for qubits or qudits, as well as linear oscillator codes which
encode logical oscillators. Unlike for qubits or oscillators, homological
quantum rotor codes allow one to encode both logical rotors and logical qudits,
depending on the homology of the underlying chain complex. In particular, such
a code based on the chain complex obtained from tessellating the real
projective plane or a M\"{o}bius strip encodes a qubit. We discuss the distance
scaling for such codes which can be more subtle than in the qubit case due to
the concept of logical operator spreading by continuous stabilizer
phase-shifts. We give constructions of homological quantum rotor codes based on
2D and 3D manifolds as well as products of chain complexes. Superconducting
devices being composed of islands with integer Cooper pair charges could form a
natural hardware platform for realizing these codes: we show that the
0-Ď€-qubit as well as Kitaev's current-mirror qubit -- also known as the
M\"{o}bius strip qubit -- are indeed small examples of such codes and discuss
possible extensions.Comment: 47 pages, 10 figures, 2 table