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

TL;DR

This work constructs explicit unitary metaplectic representations of $SL_2(\mathbb{Z}_{2^n})$ by extending the finite quantum phase space to a doubled Hilbert space and employing twisted magnetic translations on the diagonal Heisenberg–Weyl group. It provides concrete realizations of the generators $S$ and $T$, and a general formula for $U(A)$ for all $A\in SL_2(\mathbb{Z}_{2^n})$, verifying the metaplectic property in a $2^{2n}$-dimensional setting. The resulting representation is exact but reducible, avoiding reliance on quadratic characters, and aligns with Weil- representation theory while highlighting trade-offs for even moduli. The construction has practical relevance for simulating $n$-qubit dynamics and suggests applications to quantum circuits and graph-based quantum information processing, with future work on non-faithful representations and quantum learning on graphs.

Abstract

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. 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 $HW_{2^n} \otimes HW_{2^n}$. Comparisson with other approaches on this problem of the literature are discussed.