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

Unitary operations are a fundamental component of quantum algorithms, but they seem to be far more useful if given with a "quantum control" as a controlled unitary operation. However, quantum operations are not limited to unitary operations. Nevertheless, it is not a priori clear if a controlled form of these general deterministic quantum operations can be well-defined. To provide a novel tool in the toolbox for quantum programming, we propose a mathematically consistent definition of a controlled form of deterministic but non-unitary quantum operations and, more generally, of quantum combs. We propose a "neutralization" comb, which transforms a set of input quantum operations to the identity operation, and study its controlled form based on our definition. We propose two new quantum algorithms for universal controllization of divisible unitary operations utilizing the most coherently controlled neutralization combs.