arXiv subjects

Rahul Dalal

Publications and source records attributed to Rahul Dalal.

8 recordsLinked to original sources

arXiv2026

Ramanujan Complexes from Unitary Groups over Number Fields

In this article, we construct new families of Ramanujan complexes with local structure distinct from all previously known examples. Our approach is based on unitary groups over number fields, more specifically on what we call super-definite unitary groups, that is definite unitary groups that are anisotropic modulo their center at a finite place. These arise naturally as groups of units in central division algebras with involution of the second kind. Our first main result gives a general construction of infinite families of Ramanujan complexes associated with a super-definite unitary group $G$ over a totally real number field and a finite place $v_0$. The structure of the resulting complex is governed by the type of the Bruhat-Tits building at $v_0$. It includes new examples of type $A_n$ when $v_0$ is split, and novel families of type ${}^2\!A'_n$, ${}^2 \! A''_n$ (with $n$ even), $B$-$C_n$, ${}^2 \! B$-$C_n$ and $C$-$BC_n$ in the non-split case. This construction works uniformly across all ranks. Since much of the motivation for constructing expander complexes comes from computer science, we investigate the algorithmic explicitness of our construction in the latter part of the paper, and provide an example in rank 5 where it becomes fully explicit. In particular, this example yields golden gates for the real Lie group $PU(5)$.

math.NT↗

arXiv2026

Long distance quantum illumination and ranging using polarization entangled photon pairs in a lossy environment

Using polarization entangled photon pairs, we demonstrate a robust scheme for quantum illumination and ranging in a lossy environment. Entangled photon pairs are generated in a Sagnac interferometer configuration, yielding high-visibility two-photon polarization entanglement with a measured CHSH parameter of $S =2.802\pm0.002$. One of the photons from the entangled pair is retained as idler and the other one is directed into either of the two paths, namely reference and probe, of which probe is sent toward a distant object through a lossy free-space channel, and the reflected photons are collected after round-trip free-space propagation over distances approaching $1$ km. Remarkably, strong correlations are observed with CHSH values $S >2.6$ even when only a few tens of probe photons are returned, confirming the robustness of polarization entanglement under long-distance free-space propagation. This work reports the robustness of encoding photons in different basis before it is sent towards the object and recovery of polarization entanglement even after a kilometer-scale scattering from the objects, establishing a practical foundation for scalable quantum-assisted object detection and ranging.

quant-ph↗

arXiv2025

Uniform bounds and uncertainty for asymptotics of representations of $p$-adic ${\rm GL}_N$

We prove two results on the growth of dimensions of fixed vectors of representations $\pi$ of $p$-adic ${\rm GL}_N$ under principal congruence subgroups: First, a uniform bound on the growth of fixed vectors in terms of the GK-dimension $\pi$, which we extend to a uniform bound on the Harish-Chandra--Howe coefficients. Second, for $\pi$ unitary, a quantitative relationship between the GK-dimension of $\pi$ and the rate of decay of its matrix coefficients. These results are independent of one another and proved in the framework of the Langlands and Zelevinsky classifications.

math.RT↗

arXiv2025

Multi-Qubit Golden Gates

Our goal in this paper is to construct optimal topological generators for compact unitary Lie groups, extending the work of a letter of Sarnak and arXiv:1704.02106 on golden and super-golden gates to higher dimensions. To do so we consider a variant of the Sarnak--Xue Density Hypotheses in the weight aspect for definite projective unitary groups and prove it using the endoscopic classification of automorphic representations. Our main motivation is to construct efficient multi-qubit universal gate sets for quantum computers. For example, we find a set of universal gates that, for a given accuracy, can heuristically approximate arbitrary unitary operations on 2 qubits with $\approx$10 times fewer ``expensive'' $T$-type gates than the standard Clifford+$T$ set. Our framework also covers the 2-qubit Clifford+CS gate set, well-known for being particularly friendly to fault-tolerant implementation. We thereby prove tight upper bounds on the required CS count for approximations (specifically, $4.8$x fewer non-Clifford gates than Clifford+$T$).

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arXiv2024

Root Number Equidistribution for Self-Dual Automorphic Representations on $GL_N$

Let $F$ be a totally real field. We study the root numbers $\epsilon(1/2, \pi)$ of self-dual cuspidal automorphic representations $\pi$ of $\mathrm{GL}_{2N}/F$ with conductor $\mathfrak n$ and regular integral infinitesimal character $\lambda$. If $\pi$ is orthogonal, then $\epsilon(1/2, \pi)$ is known to be identically one. We show that for symplectic representations, the root numbers $\epsilon(1/2, \pi)$ equidistribute between~$\pm 1$ as $\lambda \to \infty$, provided that there exists a prime dividing $\mathfrak n$ with power $>N$.We also study conjugate self-dual representations with respect to a CM extension $E/F$, where we obtain a similar result under the assumption that $\mathfrak n$ is divisible by a large enough power of a ramified prime and provide evidence that equidistribution does not hold otherwise. In cases where there are known to be associated Galois representations, we deduce root number equidistribution results for the corresponding families of $N$-dimensional Galois representations. The proof generalizes a classical argument for the case of $\mathrm{GL}_2/\mathbb Q$ by using Arthur's trace formula and the endoscopic classification for quasisplit classical groups similarly to a previous work (arxiv:2212.12138). The main new technical difficulty is evaluating endoscopic transfers of the required test functions at central elements.

math.NT↗

arXiv2022

Statistics of Cohomological Automorphic Representations on Unitary Groups via the Endoscopic Classification

Consider the family of automorphic representations on a unitary group with cohomological factor $\pi_0$ at infinity and given split level. We compute statistics of this family as the level goes to infinity. For unramified unitary groups and a large class of $\pi_0$, we use the endoscopic classification of representations to compute the exact leading term for counts of representations and averages of Satake parameters. The bounds on our error terms are similar to previous work by Shin-Templier who studied the case of discrete series at infinity. We also prove new upper bounds for all cohomological representations. This has many corollaries: new exact asymptotics on the growth of cohomology in certain towers of locally symmetric spaces, an averaged Sato-Tate equidistribution law for spectral families with specific non-tempered cohomological components at infinity, and the Sarnak-Xue density hypothesis for cohomological representations at infinity on all unitary groups of rank $\geq 5$.

math.NT↗

arXiv2021

Counting Discrete, Level-$1$, Quaternionic Automorphic Representations on $G_2$

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm{GL}_2$. Here, we use "hyperendoscopy" techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$, eventually getting an analog of the Eichler-Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard, and Ta\"ibi to compute dimensions of spaces of level-$1$ quaternionic representations. On the way, we prove a Jacquet-Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$. The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being "regular". A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series. We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.

math.NT↗

arXiv2020

Sato-Tate Equidistribution for Families of Automorphic Representations through the Stable Trace Formula

In arXiv:1208.1945, Shin and Templier proved certain equidistribution bounds on local components of certain families of automorphic representations. We extend their weight-aspect results to families of automorphic representations where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$-packet. We do this by using a so-called "hyperendoscopy" version of the stable trace formula developed by Ferrari. The main technical difficulties are defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and bounding the values of transfers of unramified functions. We also present an extension of Arthur's simple trace formula for test functions with Euler-Poincaré component at infinity to non-cuspidal groups since it does not seem to appear elsewhere in the literature.

math.NT↗