Random quantum circuits, composed of random unitary gates, serve as canonical models of quantum dynamics. They provide critical insights into the nature of complex quantum many-body systems and have broad technological applications in quantum information. Situated at the intersection of representation theory, graph theory, stochastic processes, and functional analysis, the mathematical theory of random quantum circuits has attracted extensive study, significantly enriching the interplay between quantum information and mathematics.
In this talk, I will introduce the theoretical framework, known as the unitary design, to construct random quantum circuits. It can be seen as a special case of stochastic processes on groups, where we pay more attention to the unitary representation of the group on a quantum system and the convergence time to the desired (moments of) distribution. I will first review recent years' progresses in designing generic random circuits. Next, I will focus on unitary designs constrained by continuous and non-Abelian group symmetries, which are physically significant yet mathematically challenging. I will present our recent results on efficiently generating random circuits with and without symmetry constraints. Several illustrative examples and applications, including the dynamics of entanglement and correlation, the growth of quantum complexity, covariant random error correction codes, will be also discussed. Ref: arXiv:2411.04893, arXiv:2411.04898.
报告人简介：Zimu Li is a PhD student at Yau mathematical science center from Tsinghua University. He studies the mathematical theory of quantum information, especially on random quantum circuits and quantum error correction codes. His research works has been accepted and published in leading journals in quantum information including Phys. Rev. X: Quantum and New Journal of Physics: Quantum Information.
