Lecture Notes in Computer Science 4910

Quantum walks are quantum counterparts of random walks. In the last 5 years, they have become one of main methods of designing quantum algorithms. Quantum walk based algorithms include element distinctness, spatial search, quantum speedup of Markov chains, evaluation of Boolean formulas and search on ”glued trees” graph. In this talk, I will describe the quantum walk method for designing search algorithms and show several of its applications. 1 Quantum Algorithms: An Overview Quantum computing (and, more broadly, quantum information science) is a new area at the boundary of computer science and physics. The laws of quantum mechanics are profoundly different from conventional physics. Quantum computing studies how to use them for the purposes of computer science and information processing. The area of quantum computing was shaped by the discoveries of two major quantum algorithms in mid-1990s. The first of the two was Shor’s polynomial time quantum algorithm for factoring and discrete logarithms. Factoring and discrete logarithm are very hard number theoretic problems. The difficulty of these problems has been used to design cryptosystems (such as RSA and Diffie-Helman key exchange) for secure data transmission over an insecure network (such as Internet). The security of data transmission rests on the assumption that it is hard to factor (or find discrete logarithm of) large numbers. Until recently, this assumption was not in doubt. Mathematicians had tried to devise an efficient way of factoring large numbers for centuries, with no success. In 1994, Shor [26] discovered a fast algorithm for factoring large numbers on a quantum mechanical computer. This shook up the foundations of cryptography. If a quantum mechanical computer is built, today’s methods for secure data transmission over the Internet will become insecure. Another, equally strikingly discovery was made in 1996, by Lov Grover [18]. He invented a quantum algorithm for speeding up exhaustive search problems. Grover’s algorithm solves a generic exhaustive search problem with N possible solutions in time O( √ N). This provides a quadratic speedup for a range of search problems, from ones that are solvable in polynomial time classically to NP-complete ones. Supported by University of Latvia Grant Y2-ZP01-100. V. Geffert et al. (Eds.): SOFSEM 2008, LNCS 4910, pp. 1–4, 2008. c © Springer-Verlag Berlin Heidelberg 2008