Ramanujan quantum expanders from the Weil representation

Abstract: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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2609.38075 [quant-ph]

(or arXiv:2609.38075v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.38075 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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