We prove that any $n$-qubit unitary transformation 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 $\Omega\big(2^{n/2}\big)$ lower bound for (i) and (ii) for a certain class of implementations.

翻译：我们证明任意n量子比特酉变换可以在以下条件下实现：（i）通过查询适当的经典预言机，近似地以时间$\tilde O\big(2^{n/2}\big)$实现；以及（ii）精确地通过深度为$\tilde O\big(2^{n/2}\big)$、使用一比特和二比特门以及$2^{O(n)}$个辅助比特的线路实现。证明过程均涉及简化为Grover搜索的归约方法。其中（ii）的证明还包含一个使用一比特和二比特门（事实上，若引入扇出门和广义Toffoli门可改进为常数深度）构建任意量子态的线性深度构造方法，该方法可能具有独立研究价值。我们还针对特定实现类别证明了（i）和（ii）的下界均为$\Omega\big(2^{n/2}\big)$。