The Role of Interference and Entanglement in Quantum Computing

ii Preface Prelude The year is 1900. The world's most renowned mathematician, David Hilbert, proposes 10 challenging mathematical problems for the nascent century at the second international congress in Paris. One of these problems, the Entscheidungsproblem, asks if there exists an algorithm that decides whether a first-order statement is derivable from the axioms of first-order predicate calculus. After more than 25 years, a young mathematician from Cambridge, Alan Turing, discovers a solution. Moreover, a solution that will inspire researchers, several generations later, to invent a revolutionary computational model based on quantum mechanics. In order to solve the Entscheidungsproblem, Alan Turing formalized the notion of an algorithm. His approach considered which calculations could be performed by an idealized human computor 1 , a computor who follows simple rules and is not limited in time and space. Starting from the fundamental observations of physical reality that operations must be easily imple-mentable and only make local changes, he defined the Turing machine and formulated a thesis which is currently known as the Church-Turing thesis 2 : The class of functions computable by a Turing machine correspond exactly to the class of functions which we would naturally regard as computable by an algorithm. The Church-Turing thesis is a connection between the mathematical notion of an algorithm and the intuitive notion of what is thought of as an algorithm. The Church-Turing thesis and Turing machines allowed researchers in the '40's and 50's to create a theory of computability-what is computable and what not. When computability theory was mature, starting from the '60s, researchers started to characterize the resources needed to solve a problem, giving birth to the active research area called complexity theory. An important concept from complexity theory that recurs in this thesis is the notion of an efficient computation: an algorithm that decides every possible input in time polynomial in the size of the input. Since the invention of Turing machines, many computer scientists and mathematicians devised other computational models such as boolean circuits, random access machines,. .. and noticed that all these models have the exact same computational power as Turing machines, thereby supporting the idea of the Turing machine as a powerful and robust model of computation. Moreover, all these models were proved to be polynomially equivalent to each other and Turing machines meaning that the different computational models could efficiently simulate one another. Only probabilistic Turing machines seemed to challenge …