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arXiv · 2111.07992

Query and Depth Upper Bounds for Quantum Unitaries via Grover Search

Gregory Rosenthal
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Abstract

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 $\Omega\big(2^{n/2}\big)$ lower bound for (i) and (ii) for a certain class of implementations.

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Gregory Rosenthal. 2021-11-15. Query and Depth Upper Bounds for Quantum Unitaries via Grover Search. https://doi.org/10.22331/q-2026-06-30-2144

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