time complexity computability quantum complexity theory sorting quantum supremacy computer ion quantum algorithms
In principle, a non-quantum (classical) computer can solve the same computational problems as a quantum computer, given enough time. Quantum advantage comes in the form of time complexity rather than computability, and quantum complexity theory shows that some quantum algorithms for carefully selected tasks require exponentially fewer computational steps than the best known non-quantum algorithms. Such tasks can in theory be solved on a large-scale quantum computer whereas classical computers would not finish computations in any reasonable amount of time. However, quantum speedup is not universal or even typical across computational tasks, since basic tasks such as sorting are proven to not allow any asymptotic quantum speedup. Claims of quantum supremacy have drawn significant attention to the discipline, but are demonstrated on contrived tasks, while near-term practical use cases remain limited. 
