  Quantum algorithms profit from the interference of quantum states in an
exponentially large Hilbert space and the fact that unitary transformations on
that Hilbert space can be broken down to universal gates that act only on one
or two qubits at the same time. The former aspect renders the direct classical
simulation of quantum algorithms difficult. Here we introduce higher-order
partial derivatives of a probability distribution of particle positions as a
new object that shares these basic properties of quantum mechanical states
needed for a quantum algorithm. Discretization of the positions allows one to
represent the quantum mechanical state of $n_\text{bit}$ qubits by
$2(n_\text{bit}+1)$ classical stochastic bits. Based on this, we demonstrate
many-particle interference and representation of pure entangled quantum states
via derivatives of probability distributions and find the universal set of
stochastic maps that correspond to the quantum gates in a universal gate set.
We prove that the propagation via the stochastic map built from those universal
stochastic maps reproduces up to a prefactor exactly the evolution of the
quantum mechanical state with the corresponding quantum algorithm, leading to
an automated translation of a quantum algorithm to a stochastic classical
algorithm. We implement several well-known quantum algorithms, analyse the
scaling of the needed number of realizations with the number of qubits, and
highlight the role of destructive interference for the cost of the emulation.
Foundational questions raised by the new representation of a quantum state are
discussed.
