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