the seed mutation in direction of an exchangeable index $k\in 
[1,m]$. For each $\varepsilon\in \{ 1,-1\}$ we define a mapping $M_k:
\ZZ^n\to  \mathfrak F$ via

$$M_k(c)=\sum_{p=0}^{c_k} \binom{c_k}{p}_{q^{d_k}{2}} M(E_\varepsilon c
+\varepsilon p b^k)\ , \quad M_k(-c)=M_k(c)^{-1}\ ,$$ where we use the well-known $q$-binomial coefficients (see e.g. [@bz-qclust Equation 4.11]), and the matrix $E_{k,\varepsilon}$ defined in Section \[se:Compatible Pairs and Mut\]. Define $B_k$ to be obtained from $B$ by the standard matrix mutation in direction $k$, as in Section \[se:Cluster Algebras-def\]. One obtains the following fact.

[@bz-qclust Prop. 4.7] (a) The map $M_k$ is a toric frame, independent of the choice of sign $\varepsilon$.

\(b) The pair $(B_k,\Lambda_{M_k})$ is a quantum seed.

Now, given an [*initial quantum seed*]{} $(B, \Lambda_M) $ denote, in a slight abuse of notation, by $X_1=M(e_1),\ldots, X_r=M(e_r)$, which we refer to as the [*cluster variables*]{} associated to the quantum seed $(M,B)$. Here our nomenclature differs slightly from [@bz-qclust], since there one considers the coefficients not to be cluster variables. We now define the seed mutation $$X_k'=M(-e_k+\sum_{b_{ik}>0} b_{ik}e_i)+ M(-e_k-\sum_{b_{ik}<0} b_{ik}e_i)\ .$$

We obtain that $X_k'=M_k(e_k)$ (see [@bz-qclust Prop. 4.9]). We say that two quantum seeds $(M,B)$ and $(M',B')$ are mutation-equivalent if they can be obtained from one another by 