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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 anappropriate 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 statesusing one- and two-qubit gates (in fact, this can be improved to constant depthwith the addition of fanout and generalized Toffoli gates) which may be ofindependent interest. We also prove a matching $\Omega\big(2^{n/2}\big)$ lowerbound for (i) and (ii) for a certain class of implementations.

Query and Depth Upper Bounds for Quantum Unitaries via Grover Search