Posted

Ewin Tang, John Wright (Aug 04 2025).
Abstract: Many quantum algorithms, to compute some property of a unitary UU, require access not just to UU, but to cUcU, the unitary with a control qubit. We show that having access to cUcU does not help for a large class of quantum problems. For a quantum circuit which uses cUcU and cUcU^\dagger and outputs ψ(U)|\psi(U)\rangle, we show how to ``decontrol'' the circuit into one which uses only UU and UU^\dagger and outputs ψ(φU)|\psi(\varphi U)\rangle for a uniformly random phase φ\varphi, with a small amount of time and space overhead. When we only care about the output state up to a global phase on UU, then the decontrolled circuit suffices. Stated differently, cUcU is only helpful because it contains global phase information about UU. A version of our procedure is described in an appendix of Sheridan, Maslov, and Mosca [SMM09]. 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 UU, UU^\dagger, cUcU, and cUcU^\dagger.

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