Construction of Metaplectic Representations of $S{L}_2({\mathbb{Z}}_{2^n})$ and Twisted Magnetic Translations

Abstract

Unitary metaplectic representations of the group $S{L}_2({\mathbb{Z}}_{2^n})$ 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 $H{W}_{2^n}\otimes H{W}_{2^n}$. Comparisson with other approaches on this problem of the literature are discussed.

arXiv comment17 pages. Minor modifications

References in corpus (9)

• Theory for Equivariant Quantum Neural Networks
• On the Classification of Modular Fusion Algebras
• The Holographic Space-Time Model of Cosmology
• The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional
• Modular discretization of the AdS2/CFT1 Holography
• Holomorphic Quantization on the Torus and Finite Quantum Mechanics
• Uncertainty relation and non-dispersive states in Finite Quantum Mechanics
• Fast Quantum Maps
• Complete set of unitary irreps of Discrete Heisenberg Group $H{W}_{2^s}$