What is computable and non-computable in the quantum domain: 7 statements and 3 conjectures

Recent progress in developing computational devices based on quantum effects and demonstrations of solving various tasks using them has actualized the question of the origin of the quantum advantage. Although various attempts to quantify and characterize the nature of quantum computational advantage have been made, this question in the general context remains open: There is no universal approach that helps to define a scope of problems that quantum computers are able to speed up, theoretically and in practice. Here we review an approach to this question based on the concept of complexity and reachability of quantum states. On the one hand, the class of quantum states that is of interest for quantum computing should be complex, i.e. non-simulatable with classical computers with less than exponential resources. On the other hand, such quantum states should be reachable on a practical quantum computer. This means that a unitary corresponding to the transformation of quantum states from initial to desired can be decomposed in a sequence of single- and two-qubit gates with of no more than polynomial in the number of qubits. Our consideration paves the way towards understanding the scope of problems that can be solved by a quantum computer by formulating a sequence of statements and conjectures on various sets of quantum states.