Galois-qudit CWS code

Description

A CWS code for Galois qudits, defined using a Galois-qudit cluster state and a set of Galois-qudit \(Z\)-type Pauli strings defined by a \(q\)-ary classical code.

This entry has not yet been developed explicitly in the literature, so the formulation below is a conjectural Galois-qudit adaptation of the modular-qudit CWS constructions in Refs. [1,2]. The Galois-qudit CWS construction takes in \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \), where \(\mathcal{G}\) is a graph, and where \(\mathcal{C}\) is an \((n,K,d)_q\) \(q\)-ary code. From the graph, we form the Galois-qudit cluster state \( |\mathcal{G} \rangle \). From the \(q\)-ary code, we form Galois-qudit Pauli \(Z\)-type operators \( W_i = Z_{c_{i,1}} \otimes \cdots \otimes Z_{c_{i,n}} \), where \(c_{i,j} \) is the \(j\)-th coordinate of the \(i\)-th classical codeword. The codewords are then \( | i \rangle = W_i | \mathcal{G} \rangle \).