Chapter 5 - Quantum Algorithms and Methods

Abstract

This chapter is devoted to basic quantum algorithms and methods. The chapter starts by revisiting the quantum parallelism concept and describing its power in calculating a global property of a certain function by performing only one evaluation of that function, namely Deutsch’s and the Deutsch–Jozsa algorithms. We also describe the Bernstein–Vazirani algorithm, which is able to determine a string encoded in a function in only one computational step. Furthermore, the Grover search algorithm to perform a search for an entry in an unstructured database is described. Next, the quantum Fourier transform is described, which is a basic algorithm used in many other quantum algorithms. In addition, an algorithm to evaluate the period of a function is provided. How to can crack the Rivest–Shamir–Adleman encryption protocol is also described. Then, Shor's factorization algorithm is provided. Furthermore, Simon's algorithm is described. Next, the quantum phase estimation algorithm is described. Quantum computational complexity and Turing machine representation are discussed as well. The quantum circuits are imperfect, which prevents us from running well-known quantum algorithms using the gates-based quantum computing approach. To solve this problem, adiabatic quantum computing and variational (quantum) circuits are introduced. After summarizing the chapter, a set of problems is offered to provide a deeper understanding of the material presented in this chapter.