Basics for Algorithms in Quantum Computing
Guillermo B. Morales-Luna · AIP conference proceedings · 2006
A short introduction to Quantum Computing, regarded as a paradigm of “parallel computing based on exterior algebras”, is presented from the point of view of its main algorithms including the primitive functions, the compositional schemes and the input and output data coding. In this study the Exterior Algebra notation is used in lieu of the more conventional Dirac’s ket notation. The basic notions of Exterior Algebra and Hilbert Spaces are sketched in order to show the qubits as points in the unit circle in the two‐dimensional complex Hilbert space H1, and any word consisting of qubits as a point in the unit sphere of a external product of H1. The computing procedure is illustrated through the classical Deutsch‐Josza’s algorithm, followed by the quantum algorithm to compute the Discrete Fourier Transform in linear time and the famous polynomial‐time Shor’s Algorithm for integer factorization. Finally, basic notions in Quantum Cryptography and Quantum Communication Complexity are discussed.