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Unitary operations are a fundamental component of quantum algorithms, butthey seem to be far more useful if given with a "quantum control" as acontrolled unitary operation. However, quantum operations are not limited tounitary operations. Nevertheless, it is not a priori clear if a controlled formof these general deterministic quantum operations can be well-defined. Toprovide a novel tool in the toolbox for quantum programming, we propose amathematically consistent definition of a controlled form of deterministic butnon-unitary quantum operations and, more generally, of quantum combs. Wepropose a "neutralization" comb, which transforms a set of input quantumoperations to the identity operation, and study its controlled form based onour definition. We propose two new quantum algorithms for universalcontrollization of divisible unitary operations utilizing the most coherentlycontrolled neutralization combs.

Controlled quantum operations and combs, and their applications to universal controllization of divisible unitary operations