Advances in the development of computing devices that use quantum effects, and the demonstration of solutions with their help various problems have stimulated a new wave of interest to the problem of the nature of the "quantum computational advantage". Although various attempts for quantifying and characterizing the nature of the quantum computational advantage have been made before, in the wide context this question is open. In fact, there is no universal approach that help to define the scope of problems that quantum computers can speed up in theory and practice. In this paper, we consider an approach to this problem that is based on the concept of the complexity and reachability of quantum states. On the one hand, the class of quantum states of interest for quantum computing must be complex, that is, not amenable to simulating by classical computers using less than exponential resources. On the other hand, such quantum states must be reachable with the use of a practical quantum computer. This means that the unitary operation corresponding to the transformation of quantum states from the initial to the desired one should be decomposable into sequences of one- and two-qubit gates at most of a polynomial length with respect to the number of qubits. By formulating a number of statements and conjectures, we consider the question of describing a class of problems that can be speed up using a quantum computer.