Bosonic stabilizer code

Description

Bosonic code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting displacement operators. Displacements form the stabilizers of the code, and have continuous eigenvalues, in contrast with the discrete set of eigenvalues of qubit stabilizers. As a result, exact codewords are non-normalizable, so approximate constructions have to be considered. Stabilizer groups are any locally compact Abelian subgroups of \(\mathbb{R}^n\), can themselves contain discrete or continuous subgroups, and can admit logical qudit and/or oscillator logical subspaces.

Stabilizer codewords encoding a finite-dimensional codespace admit a discrete infinite stabilizer group and encode quantum information in a lattice. Such qudit-into-oscillator stabilizer codes are GKP and multimode GKP codes.

Stabilizer codewords encoding a logical oscillator (i.e., CV quantum information) admit either a discrete or a continuous stabilizer group. The former, called oscillator-into-oscillator GKP codes, are obtained from multimode GKP codes by removing stabilizer generators for some of the modes. The latter encode information in hyperplanes and can be defined in terms of the continuous group's Lie algebra, i.e., as the common \(0\)-eigenvalue eigenspace of mutually commuting linear combinations of oscillator position and momentum operators called nullifiers [1] or annihilators. An oscillator-into-oscillator stabilizer code encoding \(k\) logical modes into \(n\) physical modes is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance.