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

The convergence of local unitary circuit ensembles to $k$-designs (distributions that emulate the Haar measure up to $k$-th moments) 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 (completely) random circuits, the crucial situations where symmetries or conservation laws are present remain little understood and are known to pose significant challenges. We propose, for the first time, an explicit local unitary ensemble that is capable of achieving unitary $k$-designs with SU$(d)$ symmetry. To achieve this, we employ 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 develop a numerical method using the Young orthogonal form and $S_n$ branching rule to study the convergence time of CQA ensemble and provide a strong evidence for nonconstant spectral gap. Then we conjecture a convergence 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 of the potential difficulties and limitations to analyze the convergence time mathematically through classical methods that worked well for the case without symmetries including local gap threshold, martingale method, and representation theory under SU$(d)$ symmetry, suggesting that a new approach is likely needed to rigorously understand the convergence time of local random circuits with SU$(d)$ symmetry.