  We report results of a Monte Carlo simulation of the $\phi^4$ quantum chain.
In order to enhance the efficiency of the simulation we combine multigrid
simulation techniques with a refined discretization scheme. The resulting
accuracy of our data allows for a significant test of an analytical
approximation based on a variational ansatz. While the variational
approximation is well reproduced for a large range of parameters we find
significant deviations for low temperatures and large couplings.
