S Is Quantum Search Practical? Q U a N T U M C O M P U T I N G

sive speedups in computing by exploiting quantum-mechanical effects such as superposition (quantum parallelism) and entanglement.1 A quantum algorithm typically consists of applying quantum gates to quantum states, but because the input to the algorithm might be normal classical bits (or nonquantum), it only affects the selection of quantum gates. After all the gates are applied, quantum measurement is performed, producing the algorithm’s nonquantum output. Deutsch’s algorithm, for instance, solves a certain artificial problem in fewer steps than any classical (nonquantum) algorithm, and its relative speedup grows with the problem size. However, Charles Bennett and his colleagues2 have shown that quantum algorithms are unlikely to solve NP-complete problems in polynomial time, although more modest speedups remain possible. Peter Shor designed a fast (polynomial-time) quantum algorithm for number factoring—a key problem in cryptography that isn’t believed to be NP-complete. No classical polynomial-time algorithm for number factoring is known, and the security of the RSA code used on the Internet relies on this problem’s difficulty. If a large and error-tolerant quantum computer were available today, running Shor’s algorithm on it could compromise e-commerce. Lov Grover’s quantum search algorithm is also widely studied. It must compete with advanced classical search techniques in applications that use parallel processing3 or exploit problem structure, often implicitly. Despite its promise, though, it is by no means clear whether, or how soon, quantum computing methods will offer better performance in useful applications.4 Traditional complexity analysis of Grover’s algorithm doesn’t consider the complexity of processing the so-called oracle queries—that is, certain questions with yes or no answers. The query process is simply treated as a black box, thus making Grover’s algorithm appealing because it needs fewer queries than classical search. However, with sufficiently time-consuming query processing, Grover’s algorithm can become nearly as slow as a simple (exhaustive) classical search. In this article, we identify requirements for Grover’s algorithm to be useful in practice: