Tiger code[1] 

A CSS-like multi-mode bosonic non-stabilizer code that generalizes the pair-cat code and whose syndromes are linear combinations of occupation-number operators.

A tiger code is defined for a pair of integer matrices, \(G\) and \(H\), satisfying a homological constraint \(GH^{\text{T}} = 0\). Stabilizer-like operators of the code are either linear combinations of occupation-number operators defined by rows of \(H\), or products of annihilation/creation operators whose powers are defined by rows of \(G\).

The structure of the logical space is determined from the homology of the integer chain complex defined by \(G\) and \(H\). 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.

Codewords are coherent states projected into the subspace defined by the \(H\)-induced constraint, and their corresponding normalizations are Gelfand-Kapranov-Zelevinsky hypergeometric functions [2,3]. Codewords can be finitely or infinitely supported in Fock space, depending on the \(H\)-induced constraint. When written in terms of coherent states, codewords are orbits of a set of fiducial coherent states (determined by the aforementioned homology calculation) under a group of tensor-product rotations generated by \(H\). Therefore, codewords consist of continuous but compact coherent-state constellations.

Using multi-index notation, a projected coherent state can be written in two ways, \begin{align} |\boldsymbol{\alpha}\rangle_{\boldsymbol{\Delta}}^{H}&\propto\int \textnormal{d}\boldsymbol{\phi}e^{i\boldsymbol{\phi}(H\hat{\mathbf{n}}-\boldsymbol{\Delta})}|\boldsymbol{\alpha}\rangle\tag*{(1)}\\&\propto\sum_{H\mathbf{n}=\boldsymbol{\Delta}}\frac{\boldsymbol{\alpha}^{\mathbf{n}}}{\sqrt{\mathbf{n}!}}|\mathbf{n}\rangle~, \tag*{(2)}\end{align} where \(\boldsymbol{\alpha}\) is a complex vector, \(\boldsymbol{\Delta}\) is an integer vector, and \(\boldsymbol{\phi}\) is a vector of phases iterating over the elements of the group generated by \(H\). Tiger codewords are of the above form, and their phase-space values \(\boldsymbol{\alpha}\) lie on a torus embedded in the complex sphere of fixed-energy coherent coherent states, satisfying \(|\alpha_j|^2 = 1\).