This mathematically intensive chapter takes us through our first steps in the domain of quantum computation (QC) algorithms. The simplest of them is the Deutsch algorithm, which makes it possible to determine whether or not a Boolean function is constant for any input. The key result is that this QC algorithm provides the answer at once, whereas in the classical case it would take two independent calculations. I describe next the generalization of the former algorithm to n qubits, referred to as the Deutsch–Jozsa algorithm. Although they have no specific or useful applications in quantum computing, both algorithms represent a most elegant means of introducing the concept of quantum computation parallelism. I then describe two most important QC algorithms, which nicely exploit quantum parallelism. The first is the quantum Fourier transform (QFT), for which a detailed analysis of QFT circuits and quantum-gate requirements is also provided. As will be shown in the next chapter, a key application of QFT concerns the famous Shor's algorithm, which makes it possible to factor numbers into primes in terms of polynomials. The second algorithm, no less famous than Shor's, is referred to as the Grover quantum database search, whose application is the identification of database items with a quadratic gain in speed.

Deutsch algorithm

Our exploration of quantum algorithms shall begin with the solution of a very basic problem: finding whether or not a Boolean function f(x) is a constant.