We prove that any $n$-qubit unitary can be implemented (i) approximately in time $\tilde O\big(2^{n/2}\big)$ with query access to an appropriate classical oracle, and also (ii) exactly by a circuit of depth $\tilde O\big(2^{n/2}\big)$ with one- and two-qubit gates and $2^{O(n)}$ ancillae. The proofs involve similar reductions to Grover search. The proof of (ii) also involves a linear-depth construction of arbitrary quantum states using one- and two-qubit gates (in fact, this can be improved to constant depth with the addition of fanout and generalized Toffoli gates) which may be of independent interest. We also prove a matching $Ω\big(2^{n/2}\big)$ lower bound for (i) and (ii) for a certain class of implementations.

翻译：我们证明，任意n量子比特酉算子可以(i)在时间$\tilde O\big(2^{n/2}\big)$内，通过对适当经典预言机的查询访问近似实现；(ii)通过深度为$\tilde O\big(2^{n/2}\big)$、使用单量子比特门和两量子比特门以及$2^{O(n)}$个辅助量子比特的电路精确实现。证明过程均归约到Grover搜索。其中(ii)的证明还涉及使用单量子比特门和两量子比特门构建任意量子态的线性深度构造（实际上，通过增加扇出门和广义Toffoli门，该深度可改进为常数深度），该结果可能具有独立研究价值。此外，对于特定类型的实现，我们还证明了(i)和(ii)的下界为匹配的$Ω\big(2^{n/2}\big)$。