Construction of Metaplectic Representations of $S{L}_2({\mathbb{Z}}_{2^n})$ and Twisted Magnetic Translations

Unitary metaplectic representations of the group $S{L}_2({\mathbb{Z}}_{2^n})$ are necessary to describe the evolution of $2^n$-dimensional quantum systems, such as systems involving $n$ qubits. It is shown that in order for the metaplectic property to be fulfilled, an increase in the dimensionality of the involved $n$-qubit Hilbert spaces, from $2^n$ to $2^{2n}$, is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup $H{W}_{2^n}\otimes H{W}_{2^n}$. Comparisson with other approaches on this problem of the literature are discussed.