Homological number-phase code[1] 


A homological \(n\)-rotor code mapped into the Fock-state space of \(n\) oscillators by identifying non-negative rotor angular-momentum states with oscillator Fock states. The resulting oscillator code can encode logical rotors or qudits due to the presence of torsion in the chain complex defining the original rotor code.

Due to the absence of negative Fock states, a given homological rotor code first has to be rotated such that it has non-trivial support in the all-positive momentum orthant. This can be done by flipping the signs of the angular momenta of some of the rotors [1; Prop. 1]. Ideal codewords are not normalizable, and approximate versions have to be constructed.

Since the original homological rotor codes are using an extension of the qubit CSS-to-homology correspondence to rotors, the mapping into oscillators makes such homological encodings possible for the oscillator.


The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension. Products of chain complexes can also yield rotor codes.

The distances of the original homological rotor code are preserved, although the resulting number-phase code is approximately error-correcting due to the non-orthogonality of Pegg-Barnett phase states [2], which act as the angular position states in the number-phase interpretation of the oscillator.


  • Bosonic code — Homological number-phase codes are bosonic codes encoding logical qudits and/or logical rotors.


  • Homological rotor code — Homological number-phase codes are mappings of homological rotor codes into harmonic oscillators. Codewords of both codes are right eigenstates of powers of Susskind–Glogower phase operators and bosonic rotation operators, with the semigroup of the former (latter) continuous (discrete).
  • Bosonic stabilizer code — Homological number-phase codewords span the joint right eigenspace of powers of the non-unitary Susskind–Glogower phase operators and unitary bosonic rotation operators.
  • Oscillator-into-oscillator GKP code — Rotor analogues of \(k\)-into-\(n\) oscillator-into-oscillator GKP codes can be constructed by initializing \(n-k\) physical rotors in superpositions of phase states and applying a Clifford semigroup encoding circuit [1].
  • Number-phase code — Homological number-phase codes and number-phase codes are both manifestations of certain rotor codes, namely, the homological rotor codes and rotor GKP codes, respectively.


Y. Xu, Y. Wang, and V. V. Albert, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679
S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986) DOI
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Zoo Code ID: homological_number-phase

Cite as:
“Homological number-phase code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/homological_number-phase
@incollection{eczoo_homological_number-phase, title={Homological number-phase code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/homological_number-phase} }
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Cite as:

“Homological number-phase code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/homological_number-phase

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/homological_number-phase.yml.