For every odd prime power $q$ and $D=q+1$ , we construct an infinite family of Ramanujan quantum expanders of degree $D$ . The construction transfers Morgenstern's Ramanujan Cayley graphs on $\operatorname{PSL}_2(\mathbb{F}_{p})$ for $p$ which is an even power of $q$ , through the odd irreducible subrepresentation of the Weil representation of $\operatorname{SL}_2(\mathbb{F}_{p})$ . For a quantum expander of dimension $N$ , our implementation uses $O(\log^2 N)$ elementary gates, and $O(\log N)$ ancilla qudits, using a fixed finite gate set depending on $q$ . An advantage compared to the previous work of Iyer, Jain, Jordan, and Somma (FOCS 2026) is that assuming the quantum circuit is implemented exactly, we satisfy the $2\sqrt{D-1}/D$ singular value bound exactly without any additive error.For every odd prime power $q$ and $D=q+1$, we construct an infinite family of Ramanujan quantum expanders of degree $D$. The construction transfers Morgenstern's Ramanujan Cayley graphs on $\operatorname{PSL}_2(\mathbb{F}_{p})$ for $p$ which is an even power of $q$, through the odd irreducible subrepresentation of the Weil representation of $\operatorname{SL}_2(\mathbb{F}_{p})$. For a quantum expander of dimension $N$, our implementation uses $O(\log^2 N)$ elementary gates, and $O(\log N)$ ancilla qudits, using a fixed finite gate set depending on $q$. An advantage compared to the previous work of Iyer, Jain, Jordan, and Somma (FOCS 2026) is that assuming the quantum circuit is implemented exactly, we satisfy the $2\sqrt{D-1}/D$ singular value bound exactly without any additive error.

quant-pharXiv:2609.38075Sep 30, 20260 citations

Applications of the Jacobi group to Quantum Mechanics

Stefan Berceanu, Andrei H Gheorghe

Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introduced. Matrix elements of the squeezed operators, expectation values of polynomial operators in infinitesimal generators of the Jacobi group, the squeezing region and a description of Mandel's parameter are presented.Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introduced. Matrix elements of the squeezed operators, expectation values of polynomial operators in infinitesimal generators of the Jacobi group, the squeezing region and a description of Mandel's parameter are presented.

arXiv:0812.0448Dec 2, 200818 citations

Generalized Schrödinger representations. II

Annals of PhysicsSep 1, 19690 citations

Generalized schrödinger representations. II

John S. Nodvik

Annals of PhysicsOct 1, 19692 citations

For every odd prime power

q

and

D=q+1

, we construct an infinite family of Ramanujan quantum expanders of degree

D

. The construction transfers Morgenstern's Ramanujan Cayley graphs on

\operatorname{PSL}_2(\mathbb{F}_{p})

for

p

which is an even power of

q

, through the odd irreducible subrepresentation of the Weil representation of

\operatorname{SL}_2(\mathbb{F}_{p})

. For a quantum expander of dimension

N

, our implementation uses

O(\log^2 N)

elementary gates, and

O(\log N)

ancilla qudits, using a fixed finite gate set depending on

q

. An advantage compared to the previous work of Iyer, Jain, Jordan, and Somma (FOCS 2026) is that assuming the quantum circuit is implemented exactly, we satisfy the

2\sqrt{D-1}/D

singular value bound exactly without any additive error.