Query and Depth Upper Bounds for Quantum Unitaries via Grover Search

TL;DR

This paper establishes upper and lower bounds on the query complexity and circuit depth for implementing any n-qubit unitary, using Grover search techniques and circuit constructions.

Contribution

It provides the first tight bounds on the resources needed for quantum unitaries, including approximate and exact implementations, with novel circuit constructions and lower bounds.

Findings

Approximate implementation in time  O(2^{n/2})

Exact implementation with depth  O(2^{n/2})

Matching lower bounds of   2^{n/2} for certain classes

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.