Yonathan Touati

Abstract Is a quantum algorithm capable of implementing an if-clause? Given a black-box subroutine, a d-dimensional unitary U from U(d), a quantum if-clause would correspond to applying it to an input qudit if and only if the value of a control qubit is 1. It was previously shown by Thompson et al. and Araujo et al. that implementations using single access to the oracle U are impossible. Our main result is a strong generalization of this impossibility result: we prove that there is no unitary oracle algorithm implementing control_phi(U)=|0><0|\otimes I+e^{i\phi(U)}|1><1|\otimes U, for some U-dependent phase phi(U), even if allowed any finite number of calls to U and its inverse, and even if required to only approximate the desired operator control_phi(U). Even further, there is no such postselection oracle algorithm, i.e. a unitary oracle algorithm followed by a binary success/fail measurement, that reports success and implements control_phi(U) with a nonzero probability for each U in U(d). Our proof relies on topological arguments which can be viewed as a modification of the Borsuk-Ulam theorem. Combining the topological arguments with the algorithm of Dong et al. in fact leads to an interesting dichotomy: implementing control_phi(U^m) in our model is possible if and only if the integer m is a multiple of the oracle's dimension d. Our impossibility no longer holds if the model is relaxed, either by dropping the worst-case requirement that the algorithm works for all U from U(d), or, remarkably, by allowing measurements more general than a binary postselection measurement. We observe that for both relaxations, inefficient algorithms exist, and it remains open whether efficient ones do.