CSE 599d - Quantum Computing The Quantum Circuit Model and Universal Quantum Computation

So far we have talked about quantum computations involving only a few qubits. In this lecture I'd like to begin to discuss how we might scale this up to more than a few qubits and to make something resembling a valid model of computation. To do proper justice to this task, we should probably review the history of the classical theory of computation, and the struggles which the early pioneers in quantum computing went though in order to define a valid model of quantum computing. But we are lucky because a lot of the pitfalls and results have already been overcome, so we won't need to dwell on this work too much but instead get to the more important pragmatic question of what is needed for a universal quantum computer. So where to begin. The natural place to begin is probably back in 1936 with Alan Turing's seminal paper " On Computable Numbers, with an Application to the Entscheidungsproblem. " In this paper Turing defined a model of carrying out a computation which is now called a Turing machine. What is a Turing machine? A Turing machine is a machine which is designed (thought-experiment-like) to mimic a person (computer) who can change the contents of a unlimited paper tape which is divided up into cells that contain symbols based on the local information about the symbols on the tape and an internal state which the person keeps track of. To be more concrete, we can imagine a machine which consists of four main components. The first is an infinite tape which has been subdivided into cells into which symbols from some alphabet can be written and erased. Usually cells that have not been written on are assumed to be filled with no symbol. The second component of the Turing machine is a head. This is a device for reading and writing symbols onto the tape. This head will occupy only one cell of the Turing tape at a time. The head can also move one cell up or down the Turing tape. Third the Turing machine contains a little local memory cell called the state register of the Turing machine. This register will be in one of a finite set of different configurations and represents the " state " of the Turing machine. There are usually special states for the Turing machine, like a halt state which halts the action …