Computable and noncomputable in the quantum domain: statements and conjectures

Significant advances in the development of computing devices based on quantum effects and the demonstration of their use to solve various problems have rekindled interest in the nature of the “quantum computational advantage.” Although various attempts to quantify and characterize the nature of the quantum computational advantage have previously been made, this question largely remains open. Indeed, there is no universal approach that allows determining the scope of problems whose solution can be accelerated by quantum computers, in theory of in practice. In this paper, we consider 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 are of interest for quantum computing must be complex, i.e., not amenable to simulation by classical computers with less than exponential resources. On the other hand, such quantum states must be reachable on a practically feasible quantum computer. This means that the unitary operation that transforms the initial quantum state into the desired one must be decomposable into a sequence of one- and two-qubit gates of a length that is at most polynomial in the number of qubits. By formulating several statements and conjectures, we discuss the question of describing a class of problems whose solution can be accelerated by a quantum computer.