Are controlled unitaries helpful?

TL;DR

This paper demonstrates that access to controlled unitaries $cU$ offers no advantage for many quantum problems beyond global phase information, and provides a method to replace controlled unitaries with simpler operations.

Contribution

It generalizes a known decontrol procedure, showing controlled unitaries are mostly unnecessary, and explores implications for quantum algorithms and pseudorandomness.

Findings

Decontrolled circuits can simulate controlled unitaries with minimal overhead.

Controlled unitaries only provide global phase information about $U$.

The work counters previous beliefs about the necessity of controlled unitaries in quantum algorithms.

Abstract

Many quantum algorithms, to compute some property of a unitary $U$, require access not just to $U$, but to $cU$, the unitary with a control qubit. We show that having access to $cU$ does not help for a large class of quantum problems. For a quantum circuit which uses $cU$ and $cU^\dagger$ and outputs $|\psi(U)\rangle$, we show how to "decontrol" the circuit into one which uses only $U$ and $U^\dagger$ and outputs $|\psi(\varphi U)\rangle$ for a uniformly random phase $\varphi$, with a small amount of time and space overhead. When we only care about the output state up to a global phase on $U$, then the decontrolled circuit suffices. Stated differently, $cU$ is only helpful because it contains global phase information about $U$. A version of our procedure is described in an appendix of Sheridan, Maslov, and Mosca (arXiv:0810.3843). Our goal with this work is to popularize this result by generalizing it and investigating its implications, in order to counter negative results in the literature which might lead one to believe that decontrolling is not possible. As an application, we give a simple proof for the existence of unitary ensembles which are pseudorandom under access to $U$, $U^\dagger$, $cU$, and $cU^\dagger$.