Construction of Metaplectic Representations of $$SL_2(\mathbb {Z}_{2^n})$$ and Twisted Magnetic Translations

Unitary metaplectic representations of the group $$\varvec{SL}_{\textbf{2}}(\mathbb {Z}_{\textbf{2}^{\varvec{n}}})$$ are necessary to describe the time evolution of $$\textbf{2}^{\varvec{n}}$$ -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 $$\textbf{2}^{\varvec{n}}$$ to $$\textbf{2}^{\textbf{2}{\varvec{n}}}$$ , is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup $$\varvec{HW}_{\textbf{2}{\varvec{n}}} \otimes \varvec{HW}_{\textbf{2}{\varvec{n}}}$$ . Comparisson with other approaches on this problem of the literature are discussed.