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 $|ψ(U)\rangle$, we show how to "decontrol" the circuit into one which uses only $U$ and $U^\dagger$ and outputs $|ψ(\varphi U)\rangle$ 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 $cU^\dagger$.

翻译：许多量子算法为了计算酉算子 $U$ 的某些性质，不仅需要访问 $U$，还需要访问 $cU$（即带控制量子比特的酉算子）。我们证明，对于一大类量子问题，访问 $cU$ 并无助益。对于一个使用 $cU$ 和 $cU^\dagger$ 并输出 $|ψ(U)\rangle$ 的量子电路，我们展示了如何对该电路进行“去控制”处理，使其仅使用 $U$ 和 $U^\dagger$，并输出 $|ψ(\varphi U)\rangle$（其中 $\varphi$ 为均匀随机相位），且仅需少量时间和空间开销。当我们仅关心 $U$ 的全局相位无关的输出态时，去控制后的电路即已足够。换言之，$cU$ 之所以有用，仅因其携带了 $U$ 的全局相位信息。我们方法的某一版本已在 Sheridan、Maslov 和 Mosca 的论文（arXiv:0810.3843）附录中描述。本文旨在通过推广该结果并探讨其含义来宣传此结论，以反驳文献中那些可能导致人们认为去控制不可行的负面结论。作为应用，我们给出了一个简单证明，表明存在一类酉系综，在可访问 $U$、$U^\dagger$、$cU$ 及 $cU^\dagger$ 时，这些系综具有伪随机性。