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Texas A&M University
Mathematics

Events for 01/16/2025 from all calendars

Noncommutative Geometry Seminar

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Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Liyuan Chen, Harvard University

Title: Universal Circuit Set using the S3 Quantum Double

Abstract: Topological quantum computation with non-Abelian anyons offers a promising path toward fault-tolerant universal quantum computation. However, the practical realization of such a system remains challenging due to the difficulty of finding suitable topological materials. In this work, we provide a comprehensive blueprint for constructing a large-scale quantum computer based on the quantum double model $\mathcal{D}(S_3)$, a specific non-Abelian topological order. We implement logical computations using quantum circuits on qubits and qutrits, including a single non-Clifford gate, compatible with near-term quantum devices. This work bridges the gap between abstract mathematical frameworks and noise-resilient quantum computation on near-term devices. Our proposal offers a promising path to realize an anyon-based quantum computer.


Departmental Colloquia

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Time: 4:00PM - 5:00PM

Location: Bloc 117

Speaker: Adrian van Kan

Title: From numerical simulations of rotating Rayleigh-Bénard convection at very low Ekman numbers to stochastic dynamics in quasi-two-dimensional turbulence

Abstract: Rapidly rotating Rayleigh-Bénard convection (RRRBC) provides a paradigm for direct numerical simulations (DNS) of geo- and astrophysical fluid flows, but the accessible parameter space, despite great computational efforts, has remained restricted to moderately fast rotation (Ekman numbers $Ek \gtrsim 10^{-8}$), while realistic values of $Ek$ for applications are orders of magnitude smaller. Reduced equations of motion, the non-hydrostatic quasi-geostrophic equations (NHQGE) describing the leading-order behavior in the limit of rapid rotation ($Ek\to 0$) cannot capture finite rotation effects. This leaves the physically most relevant part of parameter space with small but finite $Ek$ currently inaccessible. I will describe the rescaled incompressible Navier-Stokes equations (RiNSE) [1,2] – a reformulation of the Navier-Stokes-Boussinesq equations informed by the scaling laws valid for $Ek\to 0$. I present the first fully nonlinear DNS of RRRBC at unprecedented rotation strengths down to $Ek=10^{-15}$ and below, showing numerically that the RiNSE predicts statistics which agree favorably with the NHQGE at very low $Ek$. This work opens the door to the exploration of a large region in the parameter space of rotating convection. Beyond the stiffness of the Navier-Stokes equations in the presence of a small parameter such as the Ekman number, the chaotic nature of turbulence also presents a significant challenge. The Navier-Stokes equations in two dimensions (2D) differ significantly from three dimensions (3D) due to additional conservation laws. Solving the 3D Navier-Stokes equations in a thin-layer geometry, there is a critical layer height $H$ below which rigorous bounding arguments show that 3D modes decay due to viscosity, leading to 2D flow. Close to this critical threshold, the energy contained in 3D modes exhibits highly intermittent dynamics. In the second part of this talk, motivated by numerical simulations of this phenomenon, I will describe stochastic dynamics of a single mode in the vicinity of a b