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The convergence of local unitary circuit ensembles to $k$-designs(distributions that emulate the Haar measure up to $k$-th moments) is a centralproblem in random quantum circuit models which play key roles in the study ofquantum information as well as physics. Despite the extensive study of thisproblem for Haar (completely) random circuits, the crucial situations wheresymmetries or conservation laws are present remain little understood and areknown to pose significant challenges. We propose, for the first time, anexplicit local unitary ensemble that is capable of achieving unitary$k$-designs with SU$(d)$ symmetry. To achieve this, we employ the novelOkounkov-Vershik approach to $S_n$ representation theory in quantum physics. Wedefine the Convolutional Quantum Alternating group (CQA) with the correspondingensemble generated by 4-local SU$(d)$-symmetric unitaries and prove that forall $k < n(n-3)/2$, they form SU$(d)$-symmetric $k$-designs in both exact andapproximate ways. We develop a numerical method using the Young orthogonal formand $S_n$ branching rule to study the convergence time of CQA ensemble andprovide a strong evidence for nonconstant spectral gap. Then we conjecture aconvergence time $\Theta(n^4 \log(1/\epsilon))$ of 1D CQA ensemble to$\epsilon$-approximate 2-design in contrast to its counterpart $O(n^2 \log(1/\epsilon))$ with no symmetry. We also provide comprehensive explanations ofthe potential difficulties and limitations to analyze the convergence timemathematically through classical methods that worked well for the case withoutsymmetries including local gap threshold, martingale method, and representationtheory under SU$(d)$ symmetry, suggesting that a new approach is likely neededto rigorously understand the convergence time of local random circuits withSU$(d)$ symmetry.

Designs from Local Random Quantum Circuits with SU(d) Symmetry