Parallelism of quantum computations from prequantum classical   statistical field theory (PCSFT)
  This paper is devoted to such a fundamental problem of quantum computing as quantum parallelism. It is well known that quantum parallelism is the basis of the ability of quantum computer to perform in polynomial time computations performed by classical computers for exponential time. Therefore better understanding of quantum parallelism is important both for theoretical and applied research, cf. e.g. David Deutsch \cite{DD}. We present a realistic interpretation based on recently developed prequantum classical statistical field theory (PCSFT). In the PCSFT-approach to QM quantum states (mixed as well as pure) are labels of special ensembles of classical fields. Thus e.g. a single (!) ``electron in the pure state'' $\psi$ can be identified with a special `` electron random field,'' say $\Phi_\psi(\phi).$ Quantum computer operates with such random fields. By one computational step for e.g. a Boolean function $f(x_1,...,x_n)$ the initial random field $\Phi_{\psi_0}(\phi)$ is transformed into the final random field $\Phi_{\psi_f}(\phi)$ ``containing all values'' of $f.$ This is the objective of quantum computer's ability to operate quickly with huge amounts of information -- in fact, with classical random fields. 
Andrei Khrennikov