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International Journal of Computing and Optimization

How to introduce techinical details of quantum computing in a theory of computation class: using the basic case of the Deutsch-Jozsa algorithm

Olga M. Kosheleva, Владик Крейнович

Published Jan 1, 2016• 4 citations

Abstract

Many students taking the theory of computation class have heard about quantum computing and are curious about it. However, the usual technical description of quantum computing requires a large amount of preliminary information, too much to fit into an already packed class. In this paper, we propose a way to introduce technical details of quantum computing that does not require much time – it can be described in less than an hour. As such an introduction, we use a simplified description of the basic case of one of the pioneering algorithms of quantum computing. 1 Formulation of the Pedagogical Problem Quantum computing is very promising. It is known that quantum computing is a promising direction in computing; see, e.g., [4]. Let us just give two examples: • a quantum computing algorithm proposed by Grover enables us to search in an unsorted array of size n in time √ n [2, 3]; • a quantum computing algorithm proposed by Shor enables us to factorize large integers in polynomial time [5]. Since the difficulty of eavesdropping on RSA-encrypted messages – which underlie most current encryption schemes – relies on the difficulty of factoring large integers, this will enable us to read all encrypted messages.

DOI: 10.12988/ijco.2016.648

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In this paper, we briefly review the basic concepts of quantum computation, entanglement, quantum cryptography and quantum fourier transform. Quantum algorithms like Deutsch Jozsa, Shor’s factorization and Grover’s data search are developed using fourier transform and quantum computation concepts to build quantum computers. Researchers are finding a way to build quantum computer that works more efficiently than classical computer. Among the standard well known algorithms in the field of quantum computation and communication we describe mathematically Deutsch Jozsa algorithm in detail for 2 and 3 qubits. Calculation of balanced and unbalanced states is shown in the mathematical description of the algorithm.

The Deutsch and Deutsch-Jozsa algorithms are foundational in the study of quantum computing, providing early and elegant demonstrations of quantum speedup over classical computation. This paper offers a pedagogical exposition of both algorithms, beginning with the necessary background in quantum computation and culminating in detailed, step-by-step derivations of the algorithms, complete with mathematical formulations and circuit diagrams. Simulations using quantum gates are also discussed to enhance understanding. The aim is to make these seminal algorithms accessible to readers with a basic background in linear algebra and quantum mechanics.

The basic ideas of quantum computation are introduced by a brief discussion of Bennett (1973, 1982) and Fredkin's (1982, 1997) ideas of reversible computation. After some remarks about Deutsch's (1985) pioneering work on quantum complexity and Shor's (1996) factorisation algorithm, quantum logic gates, qubits and registers are discussed. The role of quantum entanglement is stressed and Grover's (1997) quantum search algorithm described in detail. The paper ends with a review of the current experimental status of quantum computers.

Quantum computing takes fully advantage of the superposition principle to increase greatly (even exponentially) the speed of calculations, relative to the classical approach. The Deutsch-Jozsa algorithm is the simplest quantum algorithm illustrating this power. Unfortunately, the standard derivation involves several ingenious steps which usually leave students feeling that they could never have figured out the algorithm by themselves. We present here a different formulation of the problem which allows students to derive the algorithm using only basic knowledge of quantum mechanics and linear algebra.

This chapter contains an introduction to the main concepts in quantum computation and entanglement. It starts with a brief introduction to computational complexity and then introduces quantum gates and quantum networks, discussing the universality issue in quantum computation. A review on the main known quantum algorithms follows, including Deutsch's, Deutsch-Jozsa's, Grover's and Shor's algorithms. The basic concepts in the theory of quantum error correction are then reviewed. The second part of the chapter is devoted to entanglement. It starts by reminding the basic definitions of entanglement and entanglement criteria for bipartite and multipartite systems, and then discusses the role that entanglement plays in the quantum algorithms described before. The chapter ends with short descriptions of NMR quantum computing, the computational model DQC1, and one-way quantum computing.