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,对于任何一元一元变换(美元)和任何一元=2 ⁇ o(n/\log n)美元,都存在一个量子电路,用最多O(4 ⁇ n)美元的门来实施一美元美元,这与只实施一美元最坏情况单数的门数无异。我们还为量子状态和对角统一变换建立了类似的结果。我们的技术以Uhlig[Math. Notes. 1974] 的工作为基础,Uhlig[Math. Notes. 1974] 的工作证明Bolean功能的大规模生产原理相似。