Quantum computation and quantum information are of great current interest in computer science, mathematics, physical sciences and engineering. They will likely lead to a new wave of technological innovations in communication, computation and cryptography. As the theory of quantum physics is fundamentally stochastic, randomness and uncertainty are deeply rooted in quantum computation, quantum simulation and quantum information. Consequently quantum algorithms are random in nature, and quantum simulation utilizes Monte Carlo techniques extensively. Thus statistics can play an important role in quantum computation and quantum simulation, which in turn offer great potential to revolutionize computational statistics. While only pseudo-random numbers can be generated by classical computers, quantum computers are able to produce genuine random numbers; quantum computers can exponentially or quadratically speed up median evaluation, Monte Carlo integration and Markov chain simulation. This paper gives a brief review on quantum computation, quantum simulation and quantum information. We introduce the basic concepts of quantum computation and quantum simulation and present quantum algorithms that are known to be much faster than the available classic algorithms. We provide a statistical framework for the analysis of quantum algorithms and quantum simulation.

Published in at http://dx.doi.org/10.1214/11-STS378 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)