On the Computation of Boolean Functions by Quantum Branching Programs via Fingerprinting Farid Ablayev Alexander Vasiliev May 20, 2008 Abstract We develop quantum fingerprinting technique for constructing quantum branching pro- grams (QBPs), which are considered as circuits with an ability to use classical bits as control variables. We demonstrate our approach constructing optimal quantum ordered binary decision diagram (QOBDD) for MOD m and DMULT n Boolean functions. The construction of our technique also allows to extend the recent result of Ambainis and Nahimovs it is based on. In addition we show how our technique works for encoding quantum information for the equality problem in the simultaneous message passing model. 1 Introduction The implementation of a large-scale quantum computing device nowadays poses a great challenge for the engineers. At the moment the most realistic way is to construct a quantum computer of a classical device and a small quantum part. That’s why the number of qubits needed for physical implementation of an algorithm is a very important complexity measure. In this paper we present a circuit viewpoint on Quantum Branching Programs (QBPs) for which this measure explicitly comes out. Graph based and algebraic definitions of classical and quantum branching programs were ex- plored in numerous papers [NA00, SA04, AGK01, AGKMP]. We suggest that a QBP can be considered as a circuit aided with an ability to use classical bits as control variables for unitary op- erations. Thus, it is quite adequate model for describing the aforementioned “classical-quantum” computations. We develop a quantum fingerprinting technique oriented for implementation in the QBP model. In general, fingerprinting means presentation of initial object by a compact fingerprint which allows to organize space-efficient computations and reliably extract the result of computation. It is generally used in randomized and quantum algorithms for testing identity of different objects such as binary strings, polynomials, matrices and etc. by simply comparing their fingerprints (see book [MR95] for more information on the subject). The research [BCWW] of Buhrman, Cleve, Watrous, and De Wolf was the first which explicitly formulated and developed fingerprinting technique for the quantum communication model. From 2001 this paper has initiated a bunch of results for quantum communications. Implicitly, the 1 Electronic Colloquium on Computational Complexity, Report No. 59 (2008) ISSN 1433-8092