Quantum code

Quantum code

Codes for quantum communication over quantum channels

Description

Code designed for transmission of quantum and/or classical information through a quantum channel for the purposes of robust storage, communication, or sensing. Transmission can be performed with side information or entanglement.

While codewords \(c\) of an ECC are elements of some alphabet \(\Sigma\), quantum codewords are \(L^2\)-normalizable complex functions on \(\Sigma\). Put differently, the configuration space of the canonical (a.k.a. computational) basis states \(|c\rangle\) of a quantum system is the classical alphabet \(\Sigma\). The table below lists the most common alphabets used in quantum codes, along with names of the corresponding systems.

alphabet \(\Sigma\)	system \(L^2(\Sigma)\)
\(\mathbb{Z}_{2}=GF(2)\)	qubit
\(GF(q)\)	Galois qudit
\(\mathbb{Z}_{q}\)	modular qudit
\(\mathbb{R}\)	bosonic mode
\(G\)	group-valued qudit
\(\mathcal{C}\)	category-valued qudit

Table I: Table listing the most common alphabets (a.k.a. configuration spaces) used in quantum codes. Here, \(GF(q)\) is a finite field, \(G\) is a group, and \(\mathcal{C}\) is a category.