Classical Simulation Complexity of Quantum Branching Programs (Draft) Farid Ablayev ∗ Abstract We present classical simulation techniques for measure once quantum branching programs. For bounded error syntactic quantum branching program of width w that computes a function with error δ we present a classical deterministic branching program of the same length and width at most (1 + 2/(1 − 2δ)) 2w that computes the same function. Second technique is a classical stochastic simulation technique for bounded error and unbounded error quantum branching programs. Our result is that it is possible stochastically-classically simulate quantum branching programs with the same length and almost the same width, but we lost bounded error acceptance property. 1 Introduction Investigations of different aspects of quantum computations in the last decade became intensively growing area of mathematics, computer science, and physics. A good source of information on quantum computations is Nielsen’s and Chuang’s book [6]. The interest in models of quantum computation has been steadily increasing since the discovery of a polynomial time algorithm for factoring by Peter Shor [12]. During the last decade different types of quantum computation models based on Turing Machines, finite automata, circuits, and branching programs have been considered. For several of these models of computations, some examples of functions were presented for which quantum models appear to be much more efficient than their classical counterparts. Complexity of classical simulation of quantum computations for different models of computations were investigated in numerous papers [3, 7, 10, 15, 11]. Branching programs are important model of computations, because of their natural relationships to machines models (Turing machines, automata) and Circuit models. Different restricted models of branching programs are widely used for hardware verification and in numerous CAD applications. In the paper we present two classical simulation techniques for measure once quantum branching programs. Our first result is the following. For bounded error syntactic quantum branching program [1] of width w that computes a function with error δ we present a classical deterministic branching program of the same length and width at most (1 + 2/(1 − 2δ)) 2w that computes the same function. The construction of corresponding deterministic branching program is based on the following properties: 1. Quantum states are unit vectors (a set of quantum states form bounded set for || · || 2 norm). 2. Unitary transformations of quantum states preserves a distance. 3. Bounded-error acceptance criteria. ∗ Work done in part while visiting Max-Plank Institute for Mathematics Bonn in 2007 Email: ablayev@ksu.ru 1 Dagstuhl Seminar Proceedings 07411 Algebraic Methods in Computational Complexity http://drops.dagstuhl.de/opus/volltexte/2008/1310