Discrete Math. Appl., Vol. 19, No. 6, pp. 555–572 (2009) © de Gruyter 2009 DOI 10.1515/DMA.2009.037 On quantum realisation of Boolean functions by the fingerprinting technique F. M. ABLAYEV and A. V. VASILYEV Abstract — In this paper, we develop the fingerprinting technique of calculation of Boolean functions in quantum calculation models. The use of the fingerprinting technique is demonstrated on the example of calculation of the function MOD m in the class of quantum OBDD (oblivious read-once branching programs). Next, the potentialities of the fingerprinting technique are demonstrated in the example of realisation of the ‘equality’ function in the quantum model of communication with a referee (the SMP communication model) and in examples of recognition of languages in quantum automata. All given realisations of Boolean functions in various quantum calculation models (OBDD, SMP communication model, and a finite automaton) are asymptotically optimal. The authors thank Juhani Karhum¨ aki for the invitation to the University of Turku and fruitful discussions of this research. This research was supported by the University of Turku and the Russian Foundation for Basic Research, grant 08–07–00449. 1. INTRODUCTION The problem of physical realisations of a full-scale quantum computer is an open problem for the modern technology. The steady decrease of element sizes in production of electronic circuits leads to the situation where the circuit functioning is subject to laws of quantum mechanics. So, as quantum nanotechnology will evolve, the classical computers will in- clude quantum components and modules. Recently, in physical research centres intensive investigations are carried out concerning the design of transistors with account for laws of quantum mechanics. When the recent technological hardships in making quantum com- puter systems will be overcome, the quantum technologies will open possibilities to realise a massive parallel processing in computing also known as the quantum parallel computing. One of the first efficient quantum algorithms (the polynomial quantum algorithm of fac- torisation of a number [1]) demonstrates potential advantages of the quantum computation models for a number of problems as compared with the classical ones. Along with problems of physical nature in development of quantum computation mod- els, serious mathematical questions wait for their answers. An important open mathematical problem consists of describing the class of problems which can be more efficiently solved by the quantum computers than by the classical ones. Originally published in Diskretnaya Matematika (2009) 21, No. 4, 3–19 (in Russian). Received October 9, 2008. Revised December 16, 2008. Brought to you by | University of Arizona Authenticated Download Date | 6/4/15 10:03 PM