[Theory] FW: Theory Lunch 2023-04-19T17:30:00.000Z

[Christopher Kang]

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Sent: Wednesday, April 19, 2023 12:13:23 AM (UTC-06:00) Central Time (US & Canada)
To: Antares Chen <antaresc at uchicago.edu>
Cc: Christopher Kang <ctkang at uchicago.edu>
Subject: Theory Lunch 2023-04-19T17:30:00.000Z


Today's Theory Lunch talk:

Han Zheng (University of Chicago): On Random Quantum Circuits with Continuous Symmetry

https://uchicago.zoom.us/j/91616319229?pwd=dDdXQnFXeGNubFRkZy9hTDQrcWlXdz09<https://urldefense.com/v3/__https://uchicago.zoom.us/j/91616319229?pwd=dDdXQnFXeGNubFRkZy9hTDQrcWlXdz09__;!!BpyFHLRN4TMTrA!-McACiF0h1A-T3A0I0E66jX96qJUlpN3qj55vJI5pqcNtUtTyAcXHCmcP90UZe-55vr0xIy8e6MoMYiyaXNpHsEWM8Fuj3tzVGk$>

Description: We propose, for the first time, an explicit unitary ensemble that is capable of achieving unitary $k$-design with SU($d$) symmetry by Schur-Weyl duality connecting both SU$(d)$ and $S_n$ actions on qudits. Based on which, we further explore the potential of using $S_n$ representation theory in quantum physics, during which we define the Convolutional Quantum Alternating group (CQA) with CQA ensemble generated by 4-local SU$(d)$ symmetric unitaries and prove that for all $k < n$, they form SU$(d)$ symmetric $k$-design in exact and approximate senses respectively. We rigorously verify that it is impossible for ensembles with unitaries of constant locality to achieve this task for arbitrary large $k$, in contrast with the common case where 2-local unitaries are quite sufficient. We device a numerical method using Young orthogonal form and $S_n$ branching rule to show a convergence rate $\Omega(n^4 \log \frac{1}{\epsilon})$ of 1D CQA ensemble to $\epsilon$-approximate 2-design, with which we explain the potential difficulties and limitations to analyze the rate mathematically with classical methods like local gap threshold, martingale method and using all-to-all interaction random circuits with the presence of SU$(d)$ symmetry.

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