  Quantum finite automata derive their strength by exploiting interference in
complex valued probability amplitudes. Of particular interest is the 2-way
model of Ambainis and Watrous that has both quantum and classical states
(2QCFA) [A. Ambainis and J. Watrous, Two-way finite automata with quantum and
classical state, Theoretical Computer Science, 287(1), pp. 299-311, 2002],
since it combines the advantage of the power of interference in a
constant-sized quantum system with a 2-way head.
  This paper is a step towards finding the least powerful model which is purely
classical and can mimic the dynamics of quantum phase. We consider weighted
automata with the Cortes-Mohri definition of language recognition [C. Cortes
and M. Mohri, Context-Free Recognition with Weighted Automata, Grammars 3(2/3),
pp. 133-150, 2000] as a candidate model for simulating 2QCFA.
  Given any 2QCFA that (i) uses the accept-reject-continue observable, (ii)
recognizes a language with one-sided error and (iii) the entries of whose
unitary matrices are algebraic complex numbers, we show a method of
constructing a weighted automaton over $\mathbb{C}$ that simulates it
efficiently.
