  We derive rank bounds on the quantized tensor train (QTT) compressed
approximation of singularly perturbed reaction diffusion partial differential
equations (PDEs) in one dimension. Specifically, we show that, independently of
the scale of the singular perturbation parameter, a numerical solution with
accuracy $0<\epsilon<1$ can be represented in QTT format with a number of
parameters that depends only polylogarithmically on $\epsilon$. In other words,
QTT compressed solutions converge exponentially to the exact solution, with
respect to a root of the number of parameters. We also verify the rank bound
estimates numerically, and overcome known stability issues of the QTT based
solution of PDEs by adapting a preconditioning strategy to obtain stable
schemes at all scales. We find, therefore, that the QTT based strategy is a
rapidly converging algorithm for the solution of singularly perturbed PDEs,
which does not require prior knowledge on the scale of the singular
perturbation and on the shape of the boundary layers.
