Reducing quantum computations to elementary unitary operations

27 Quantum computing has enormous potential for introducing fundamentally new capabilities to computational science and engineering, primarily through exponential parallelism.1,2 One of the many challenges in building practical quantum computers is to reduce a general quantum computation to some set of elementary operations that simple quantum devices can implement. By analogy, in the case of classical computing, it is well known that the elementary operations AND, OR, and NOT are sufficient to implement any finite classical computation. In fact, the single NOR operation by itself is sufficient.3 Over the past 10 years, several researchers have addressed the question of reducing quantum computations to elementary operations.4–6 My goal here is to collect those results, simplify them, and present the basic ideas in terms of traditional linear algebraic operations.7 Some background