Basic Quantum Algorithms

Abstract

Quantum computing is evolving so rapidly that it forces us to revisit, rewrite, and update the foundations of the theory. \emph{Basic Quantum Algorithms} revisits the earliest quantum algorithms. The journey began in 1985 with Deutsch attempting to evaluate a function at two domain points simultaneously. Then, in 1992, Deutsch and Jozsa created a quantum algorithm that determines whether a Boolean function is constant or balanced. The following year, Bernstein and Vazirani realized that essentially the same algorithm could be used to identify a specific Boolean function within a set of linear Boolean functions. In 1994, Simon introduced a novel quantum algorithm that determines whether a function is one-to-one or two-to-one exponentially faster than any classical algorithm for the same problem. That same year, Shor developed two groundbreaking quantum algorithms for integer factoring and calculating discrete logarithms, posing a threat to widely used cryptographic methods. In 1995, Kitaev proposed an alternative formulation based on phase estimation that proved valuable in numerous applications. The following year, Grover devised a quantum search algorithm that is quadratically faster than its classical counterpart. More than a decade later, Harrow, Hassidim, and Lloyd proposed a quantum algorithm for solving systems of linear equations, now known as the HHL algorithm. With an emphasis on the circuit model, this work provides a detailed description of all these remarkable algorithms.