There is considerable controversy about whether physical realizations of quantum computation models can be built robustly enough to solve problems unsolvable classically

Among these are Grover’s algorithm (Grover 1996) for searching an unsorted database (which runs in time \(O(n^{1/2})\), whereas the best possible classical algorithm is \(O(n)\)) and Shor’s algorithm (Shor 1999) for integer factorization (which runs in \(O(\log_2(n)^3)\), whereas the best known classical algorithm is \(O(2^{\log_2(\log_2(n))^{1/3})}\)). Since it can be shown that quantum models can simulate models such as the classical Turing machine, \(\textbf{BQP}\) contains \(\textbf{P}\) and