Unitary Evolution on a Phase Space with Power of Two Discretization

Stam Nicolis, Emmanuel G. Floratos

Published May 3, 2018• 0 citations

Abstract

We construct quantum evolution operators on the space of states, that realize the metaplectic representation of the modular group SL(2,Z_2^n). This representation acts in a natural way on the coordinates of the non-commutative 2-torus and thus is relevant for noncommutative field theories as well as theories of quantum space-time. The larger class of operators, thus defined, may be useful for the more efficient realization of new quantum algorithms.

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Unitary Evolution on a Phase Space with Power of Two Discretization

Stam Nicolis, Emmanuel G. Floratos

Published May 3, 2018• 0 citations

OpenAlex

Abstract

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arXiv:hep-lat/0510043Proceedings of XXIIIrd International Symposium on Lattice Field Theory — PoS(LAT2005)Dec 14, 20051 citation

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arXiv:hep-th/0006006Jun 1, 20003 citations

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

Emmanuel G. Floratos, Kimon Manolas, I. Tsohantjis

Unitary metaplectic representations of the group $$\varvec{SL}_{\textbf{2}}(\mathbb {Z}_{\textbf{2}^{\varvec{n}}})$$ are necessary to describe the time evolution of $$\textbf{2}^{\varvec{n}}$$ -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 $$\textbf{2}^{\varvec{n}}$$ to $$\textbf{2}^{\textbf{2}{\varvec{n}}}$$ , is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup $$\varvec{HW}_{\textbf{2}{\varvec{n}}} \otimes \varvec{HW}_{\textbf{2}{\varvec{n}}}$$ . Comparisson with other approaches on this problem of the literature are discussed.Unitary metaplectic representations of the group $$\varvec{SL}_{\textbf{2}}(\mathbb {Z}_{\textbf{2}^{\varvec{n}}})$$ are necessary to describe the time evolution of $$\textbf{2}^{\varvec{n}}$$ -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 $$\textbf{2}^{\varvec{n}}$$ to $$\textbf{2}^{\textbf{2}{\varvec{n}}}$$ , is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup $$\varvec{HW}_{\textbf{2}{\varvec{n}}} \otimes \varvec{HW}_{\textbf{2}{\varvec{n}}}$$ . Comparisson with other approaches on this problem of the literature are discussed.

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We develop a systematic analysis of the metaplectic semigroup $\mathrm{Mp}_+(d,\mathbb{C})$ associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by Hörmander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group $\mathrm{Mp}(d,\mathbb{R})$ to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.We develop a systematic analysis of the metaplectic semigroup $\mathrm{Mp}_+(d,\mathbb{C})$ associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by Hörmander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group $\mathrm{Mp}(d,\mathbb{R})$ to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.

math.APCross-listedarXiv:2601.22252Feb 2, 2026

Construction of Metaplectic Representations of

SL_2(\mathbb {Z}_{2^n})

and Twisted Magnetic Translations

We develop a systematic analysis of the metaplectic semigroup

\mathrm{Mp}_+(d,\mathbb{C})

associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by Hörmander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group

\mathrm{Mp}(d,\mathbb{R})

to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.