Homological rotor code[1] 


A homological quantum rotor code is an extension of analog stabilizer codes to rotors. The code is stabilized by a continuous group of rotor \(X\)-type and \(Z\)-type generalized Pauli operators. Codes are formulated using an extension of the qubit CSS-to-homology correspondence to rotors. 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, i.e., encoding logical qudits instead of only logical rotors. Such finite-dimensional encodings are not possible with analog stabilizer codes.

A homological rotor code encoding \(k\) logical rotors and a \(q\)-dimensional logical qudit is denoted as \([[n,(k,q)]]_{\mathbb{Z}}\) or \([[n,(k,q),(d_X,\delta_Z)]]_{\mathbb{Z}}\), where \(d_X\) and \(\delta_Z\) are the code's \(X\) and \(Z\) distances, respectively. The subscript \(\mathbb{Z}\) refers to the label used for the rotor's angular momentum, but shifts in the dual angular position degree of freedom are also used to construct stabilizers (the altenative subscript \(U(1)\) is used in some cases).

The stabilizer group is defined using two integer matrices \(H_X\in\mathbb{Z}^{r_X\times n}\) and \(H_Z\in\mathbb{Z}^{r_Z\times n}\) which are such that \begin{align} H_XH_Z^T = 0.\label{eq:commutation} \tag*{(1)}\end{align} The stabilizer is then defined as \begin{align} \mathsf{S}=\left\langle e^{-i\boldsymbol{s}H_{X}\cdot\hat{\boldsymbol{L}}}e^{i\boldsymbol{\varphi}H_{Z}\cdot\hat{\boldsymbol{\phi}}}\middle\vert\forall\boldsymbol{s}\in\mathbb{Z}^{r_{X}},\forall\boldsymbol{\varphi}\in U(1)^{r_{Z}}\right\rangle .\label{eq:stabilizer} \tag*{(2)}\end{align} The condition (1) ensures that \(\mathsf{S}\) has a common +1 eigenspace.

As with CSS codes, there is a natural connection to a length-3 integer chain complexes, \begin{align} \mathcal{A}:~\mathbb{Z}^{r_X} \xrightarrow{H_X} \mathbb{Z}^n \xrightarrow{H_Z^T} \mathbb{Z}^{r_Z}~, \tag*{(3)}\end{align} whose middle homology group describes the logical \(X\) operators fo the code. The logical \(Z\) operators are defined by the middle cohomology group where the cohomology is taken with phase coefficients, \(\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}\), \begin{align} \mathcal{A}^*:~\mathbb{T}^{r_X} \xleftarrow{H_X^T} \mathbb{T}^n \xleftarrow{H_Z} \mathbb{T}^{r_Z}. \tag*{(4)}\end{align}

The logical subspace can contain logical rotors as well as logical qudits. The former correspond to the so called free part of the homology group while the latter correspond to the torsion part, \begin{align} H_1(\mathcal{A},\mathbb{Z}) = \mathbb{Z}^{k^\prime}\oplus\mathbb{Z}_{d_1}\oplus\cdots\oplus\mathbb{Z}_{d_{k^{\prime\prime}}}. \tag*{(5)}\end{align}


The \(X\) distance \(d_X\) of a code is the (slightly generalized notion of) weight of the smallest logical operator constructed out of angular position shifts. The \(Z\) distance \(\delta_Z\) depends on whether or not the code encodes logical rotors, but a similar notion exists in the case of only a logical qudit encoding. One can extend the idea of disjointness [2] to rotors to obtain distance bounds [1].

Transversal Gates

All generalized Pauli gates are realized transversally.


Some logical gates come from the rotor Clifford group [3].


A Sage notebook to look at small examples can be found in github.com/cianibegood/quantum-rotor-codes.




  • Homological number-phase 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).
  • Qudit cubic code — The qudit cubic code can be generalized to rotors [4,5].


C. Vuillot, A. Ciani, and B. M. Terhal, “Homological Quantum Rotor Codes: Logical Qubits from Torsion”, (2023) arXiv:2303.13723
T. Jochym-O’Connor, A. Kubica, and T. J. Yoder, “Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates”, Physical Review X 8, (2018) arXiv:1710.07256 DOI
Y. Xu, Y. Wang, and V. V. Albert, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679
J. Haah, Two generalizations of the cubic code model, KITP Conference: Frontiers of Quantum Information Physics, UCSB, Santa Barbara, CA.
V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
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Zoo Code ID: homological_rotor

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/rotors/stabilizer/css/homological_rotor.yml.