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

TL;DR

This paper constructs explicit unitary metaplectic representations of the group $SL_2(Z_{2^n})$ relevant for quantum systems with $n$ qubits, revealing the need for higher-dimensional Hilbert spaces and using magnetic translations.

Contribution

It provides a new construction of metaplectic representations based on magnetic translations, expanding understanding of quantum evolutions in multi-qubit systems.

Findings

Metaplectic representations require increasing Hilbert space dimension from $2^n$ to $2^{2n}$.

Constructed explicit matrix form of representations using magnetic translations.

Compared new approach with existing literature methods.

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.