内容：

The convergence of local unitary circuit ensembles to k-designs is a central problem in random quantum circuit models which play key roles in the study of quantum information as well as physics. Despite the extensive study of this problem for Haar random circuits, the crucial situations where symmetries or conservation laws are present remain little understood and are known to pose significant challenges. We introduce in this talk a series of our recent works on local random quantum circuits and k-designs under SU(d) symmetry. We propose, for the first time, an explicit local unitary ensemble that is capable of achieving unitary k-designs with SU(d) symmetry using the novel Okounkov-Vershik approach to S_n representation theory in quantum physics. We define the Convolutional Quantum Alternating group (CQA) with the corresponding ensemble generated by 4-local SU(d)-symmetric unitaries and prove that for all k < n(n-3)/2, they form SU(d)-symmetric k-designs in both exact and approximate ways. We prove the polynomial convergence time for CQA ensemble to approximate 2-design using representation theory and Markov mixing theory and, we also provide explanations why it is difficult to analyze the convergence time mathematically through classical methods that worked well for the case without symmetries like local gap threshold and martingale method. If time permits, we will explicate how to apply these results to study quantum circuits complexity growth, information scrambling with non-Abelian conserved quantities, covariant quantum error correcting random codes, and geometric quantum machine learning.

Refs: arXiv:2309.08155 and arXiv:2309.16556.

人物介绍：

Zimu Li is a visiting student at Yau Mathematical Sciences Center working with Prof. Zi-Wen Liu. His main research interests are quantum information theory problems like random circuits, quantum error correction codes, quantum system with symmetries, and theoretical machine learning problems like equivariant neural network, efficient learning of molecular dynamics. He focuses on applying mathematical methods like representation theory, stochastic process and algebraic graph theory to these problems.