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We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order t. The spectral gap of this random walk is shown to be Ω(1/t), which coincides with the best previously known bound for a random walk on the permutation group on {0, 1}^n. This implies that the walk gives an ε-approximate unitary t-design. Our simple proof uses quadratic Casimir operators of Lie algebras.
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null (Ed.)Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the ``mixed'' phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems — i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears — purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.more » « less
