On Complexity of Quantum Branching Programs Computing Equality-like Boolean Functions Farid Ablayev Airat Khasianov Alexander Vasiliev January 25, 2010 Abstract We consider the Hidden Subgroup, and Equality -related problems in the context of quan- tum Ordered Binary Decision Diagrams. For the decision versions of considered problems we show polynomial upper bounds in terms of quantum OBDD width. We apply a new modi- fication of the fingerprinting technique and present the algorithms in circuit notation. Our algorithms require at most logarithmic number of qubits. 1 Introduction Considering one-way quantum finite automata, Ambainis and Freivalds (see [AF98]) suggested that first quantum-mechanical computers would consist of a comparatively simple quantum-mechanical part connected to a classical computer. In this paper we consider another restricted model of quantum-classical computation referred to as oblivious Ordered Read-Once Quantum Branching Programs. It is also known as non-uniform automata. Two models of quantum branching programs were introduced by Ablayev, Gainutdinova, Karpin- ski [AGK01] (leveled programs ), and by Nakanishi, Hamaguchi, Kashiwabara [NHK00] (non-leveled programs ). Later it was shown by Sauerhoff [SS04] that these two models are polynomially equiv- alent. In this paper we use the generalized fingerprinting technique introduced in [AV08]. The basic ideas of this approach date back to 1979 [Fre79] (see also [MR95]). It was later successfully applied in the quantum automata setting by Ambainis and Freivald in 1998 [AF98] (later improved in [AN08]). Subsequently, the same technique was adapted for the quantum branching programs by Ablayev, Gainutdinova and Karpinski in 2001 [AGK01], and was later generalized in [AV08]. The hidden subgroup problem [ME99], [Høy97] is an important computational problem that has factoring and discrete logarithm as its special cases. Subsequently, an efficient algorithm for the hidden subgroup problem implies efficient solutions for both the period finding problem, and original Simon problem. We show refined proof of the linear upper bound for the Hidden Subgroup Problem [KH06]. We prove linear upper bounds for Equality, Palindrome and boolean variant of Periodicity and Semi-Simon problems. Our upper bounds hold for arbitrary ordering of the input variables and were initially presented in [KH05], and can also be found in [AKK]. 1 ISSN 1433-8092