# Homological rotor code[1]

## Description

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} Stabilizer generator matrices equivalent under CSS rotor Clifford group transformations are classified by distinct Smith normal forms [1,2]

## Protection

## Transversal Gates

## Gates

## Notes

## Parents

- Rotor stabilizer code
- Generalized homological-product CSS code — Homological rotor 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. Products of chain complexes can also yield rotor codes.

## Children

- Kitaev current-mirror qubit code
- Three-rotor code — Taking \(H_X=\begin{pmatrix}-3 & 1 & 2\end{pmatrix}\) and \(H_Z=\begin{pmatrix}4&6&3\end{pmatrix}\) yields the three-rotor code.
- Four-rotor code
- Zero-pi qubit code

## Cousins

- 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].

## References

- [1]
- C. Vuillot, A. Ciani, and B. M. Terhal, “Homological Quantum Rotor Codes: Logical Qubits from Torsion”, (2023) arXiv:2303.13723
- [2]
- Y. Xu, Y. Wang, and V. V. Albert, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679
- [3]
- 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
- [4]
- J. Haah, Two generalizations of the cubic code model, KITP Conference: Frontiers of Quantum Information Physics, UCSB, Santa Barbara, CA.
- [5]
- 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

## Page edit log

- Victor V. Albert (2023-04-12) — most recent
- Christophe Vuillot (2023-03-28)

## Cite as:

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