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

Unitary metaplectic representations of the group $SL_2(\mathbb{Z}_{2^n})$ are necessary to describe the evolution of $2^n$-dimensional quantum systems, such as systems involving $n$ qubits. We construct the general matrix form of such representations, together with appropriately defined magnetic translations based on the diagonal subgroup of the tensor product of the finite Heisenberg group $HW_{2^n} \otimes HW_{2^n}$. 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.