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arXiv · 2505.20983

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

Emmanuel Floratos·Kimon Manolas·Ioannis Tsohantjis
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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.

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Emmanuel Floratos, Kimon Manolas, Ioannis Tsohantjis. 2025-05-27. Construction of Metaplectic Representations of $SL_2(\mathbb{Z}_{2^n})$ and Twisted Magnetic Translations. https://doi.org/10.1007/s10773-025-06103-5

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