An ε-approximate unitary k-design on n qubits can be implemented in depth O(log k · log log(nk/ε)) using long-range two-qubit gates, with near-optimal ancilla and randomness costs.
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Unitary designs in nearly optimal depth
An ε-approximate unitary k-design on n qubits can be implemented in depth O(log k · log log(nk/ε)) using long-range two-qubit gates, with near-optimal ancilla and randomness costs.