Quantum Mass Production Theorems
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We prove that for any n-qubit unitary transformation U and for any r = 2^o(n / log n), there exists a quantum circuit to implement U^⊗ r with at most O(4^n) gates. This asymptotically equals the number of gates needed to implement just a single copy of a worst-case U. We also establish analogous results for quantum states and diagonal unitary transformations. Our techniques are based on the work of Uhlig [Math. Notes 1974], who proved a similar mass production theorem for Boolean functions.
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