The idea of quantum computation, in the algorithmic sense, originated from the suggestion by Feynman (1982) that a computer based on the principles of quantum mechanics might be capable of efficiently simulating quantum systems of interest to physicists; such simulation seems to be very difficult with classical computers. Feynman's suggestion was followed up by Deutsch (1985), who introduced the notion of the quantum Turing machine and investigated the possible computational power of physically realizable computers. He showed that a specific problem, now known as Deutsch's problem, can be solved more efficiently by a quantum algorithm than by a classical algorithm. Several years later, Shor (1994) discovered efficient quantum algorithms for two important practical problems – integer factorization and the “discrete logarithm” problem – and shortly afterwards, Grover (1996) discovered an efficient quantum algorithm for unstructured searching. Since then, quantum algorithmics and quantum complexity theory have developed into substantial and active research fields.

Meanwhile, the principles of quantum mechanics were being used as the foundation for a new approach to cryptography. Bennett and Brassard (1984) defined a protocol for key distribution whose security is guaranteed by the laws of quantum theory. Their system built on earlier work by Wiesner (1983), which remained unpublished until several years after its conception. We regard quantum cryptography as an aspect of quantum computation, in particular distributed quantum computation; alternatively, both quantum algorithmics and quantum cryptography can be viewed as branches of quantum information processing.