Query and Depth Upper Bounds for Quantum Unitaries via Grover Search
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We prove that any n-qubit unitary can be implemented (i) approximately in time Õ(2^n/2) with query access to an appropriate classical oracle, and also (ii) exactly by a circuit of depth Õ(2^n/2) with one- and two-qubit gates and 2^O(n) ancillae. The proofs of (i) and (ii) 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(2^n/2) lower bound for (i) and (ii) for a certain class of implementations.
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