Query and Depth Upper Bounds for Quantum Unitaries via Grover Search

• URL: http://arxiv.org/abs/2111.07992v4
• Date: Mon, 10 Jul 2023 15:10:36 GMT
• Title: Query and Depth Upper Bounds for Quantum Unitaries via Grover Search
• Authors: Gregory Rosenthal
• Abstract summary: We prove that any $n$-qubit unitary transformation can be implemented approximately in time. We also prove a matching $Omegabig (2n/2big)$ lower bound for (i) and (ii) for a certain class of implementations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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