The quantum condition space and a classification of quantum circuits with qubit functional configurations*

In this work we first propose to exploit the fundamental properties of quantum physics to evaluate the probability of events with projection measurements. Next, to study what events can be specified by quantum methods, we introduce the concept of the condition space, which is found to be the dual space of the classical outcome space of bit strings. Just like the classical outcome space generates the quantum state space, the condition space generates the quantum condition space that offers a novel perspective of understanding quantum states with the duality picture. In addition, the quantum conditions have physical meanings and realizations of their own and thus may be studied for purposes beyond the original motivation of characterizing events for probability evaluation. An immediate application of the theory is to use the relation between the condition space and quantum circuits to systematically study how state vector entries are collectively modified by quantum gates: this leads to a classification of quantum circuits into types defined by qubit functional configurations.