  Physical systems are often simulated using a stochastic computation where
different final states result from identical initial states. Here, we derive
the minimum energy cost of simulating a complex data set of a general physical
system with a stochastic computation. We show that the cost is proportional to
the difference between two information-theoretic measures of complexity of the
data - the statistical complexity and the predictive information. We derive the
difference as the amount of information erased during the computation. Finally,
we illustrate the physics of information by implementing the stochastic
computation as a Gedankenexperiment of a Szilard-type engine. The results
create a new link between thermodynamics, information theory, and complexity.
