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 $U^{\otimes r}$ 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.

翻译：我们证明，对于任意$n$量子比特酉变换$U$和任意$r = 2^{o(n / \log n)}$，存在一个量子电路可用至多$O(4^n)$个门实现$U^{\otimes r}$。这一门数渐近等于实现单个最坏情形$U$所需门数。我们还针对量子态和对角酉变换建立了类似结果。我们的方法基于Uhlig [Math. Notes 1974]的工作，他证明了布尔函数类似的批量生产定理。