Quantum spherical code (QSC)[1] 

Code whose codewords are superpositions of points on an \(n\)-dimensional real or complex sphere. Such codes can in principle be defined on any configuration space housing a sphere, but the focus of this entry is on QSCs constructed out of coherent-state constellations.

More technically, a QSC is a collection \(\{\mathcal{C}_k\}_{k=1}^K\) of logical constellations, each of which yields a codeword by taking a quantum superposition of all points \(\mathbf{x}\in \mathcal{C}_k\). Taken together, the logical constellations yield the code constellation, \(\mathcal{C}=\bigcup_{k=1}^{K}\mathcal{C}_{k}\).

Codewords of coherent-state QSCs of uniform superposition are defined as \begin{align} |\mathcal{C}_{k}\rangle\sim\frac{1}{\sqrt{|{\mathcal{C}}_{k}|}}\sum_{\boldsymbol{\alpha}\in\mathcal{C}_{k}}|\sqrt{\bar{N}}\boldsymbol{\alpha}\rangle~, \tag*{(1)}\end{align} where \( |\boldsymbol{\alpha} \rangle = |\alpha_1,\alpha_2,...\alpha_n \rangle \) is an \(n\)-mode coherent state. This asymptotic expression is valid in the limit of large energy \(\bar{N}\to\infty\).

Coherent-state QSCs on \(n\) modes are denoted by \(((n,K,d_E,\langle t_{\downarrow},d_{\updownarrow},d_{\downarrow}\rangle))\), where \(K\) is codespace dimension, \(d_E\) is the squared minimum distance, i.e., the smallest Euclidean distance between pairs of distinct points across all codewords, and \( t_{\downarrow},d_{\updownarrow},d_{\downarrow} \) are the number of correctable losses (plus 1), the degree distance, and the number of detectable losses (plus 1), respectively.