  We present a framework using the Quantized Tensor Train (QTT) decomposition
to accurately and efficiently solve volume and boundary integral equations in
three dimensions. We describe how the QTT decomposition can be used as a
hierarchical compression and inversion scheme for matrices arising from the
discretization of integral equations. For a broad range of problems,
computational and storage costs of the inversion scheme are extremely modest
$O(\log N)$ and once the inverse is computed, it can be applied in $O(N \log
N)$.
  We analyze the QTT ranks for hierarchically low rank matrices and discuss its
relationship to commonly used hierarchical compression techniques such as FMM
and HSS. We prove that the QTT ranks are bounded for translation-invariant
systems and argue that this behavior extends to non-translation invariant
volume and boundary integrals.
  For volume integrals, the QTT decomposition provides an efficient direct
solver requiring significantly less memory compared to other fast direct
solvers. We present results demonstrating the remarkable performance of the
QTT-based solver when applied to both translation and non-translation invariant
volume integrals in 3D.
  For boundary integral equations, we demonstrate that using a QTT
decomposition to construct preconditioners for a Krylov subspace method leads
to an efficient and robust solver with a small memory footprint. We test the
QTT preconditioners in the iterative solution of an exterior elliptic boundary
value problem (Laplace) formulated as a boundary integral equation in complex,
multiply connected geometries.
