Algorithm for Quantum Simulation

Quantum computers are touted as revolutionary systems for tractably factorizing integers and solving discrete logarithms, but, in 1982, Feynman made the first proposal of a quantum computer application. He suggested that a quantum computer could efficiently simulate any local quantum system; moreover he claimed that such systems cannot in general be efficiently simulated by a (classical) Turing computer [1]. Although Feynman’s intuition is legendary, the implications of his conjecture are profound, as proving a quantum computer is strictly superior to a classical computer would also prove that the complexity classes P and PSPACE are not equal. In 1985, Deutsch generalized the Turing machine to a quantum version [2]. This introduction of a quantum computer opened the doors to studying the computation power of quantum systems in much the same way that decades of computer science research had explored the prowess and limits of classical mechanical computational devices. The theoretical side lay dormant, however, until Lloyd’s analysis of Feynman’s conjecture in 1996 [3]. Lloyd’s approach to Feynman’s conjecture was to discretize the continuous-time t evolution in terms of steps of size t/r for r the number of intervals into which the total time is divided: