We present a divide–et–impera extension of the (Quantics) Tensor Cross
Interpolation (TCI) algorithm that adaptively partitions a high
dimensional tensor into a collection of low-rank tensor train (TT) patches.
Each patch is compressed with an explicit bond-dimension cap χpatch,
that triggers finer partitioning of the configuration space wherever the
input tensor has more interesting features (higher local rank). The local
cap χpatch not only reduces the memory footprint of the tensor-train
representation of functions with sharply local features, but also tames the
O(χ4) cost of MPO-MPO contractions by decomposing the global
product into many rank-≤χpatch sub-contractions; in this context, the
choice of MPO patching scheme is essential, as it can markedly
enhance—or, if poorly chosen, limit—the overall efficiency of patched
contractions.
We derive closed-form bounds that relate χpatch and the patch count
Npatch to the memory and run-time advantage over a monolithic TCI or
MPO contraction, and identify an “over-patching” regime that arises if
the cap is chosen too small. The theoretical estimates are validated by
comprehensive benchmarks and the advantage is tested on three
notorious bottlenecks ofmany-bodyphysics related to the Hubbard
model: (i) the approximaton of a two-dimensional Matsubara Green’s
function, (ii) the computation of the bare susceptibility χ0(q,iω) (bubble
diagram), and (iii) vertex contractions entering the Bethe-Salpeter
equation for the single-impurity Anderson model. In all cases the patched
strategy yields significant memory savings together with speed-ups of
nearly an order of magnitude, enabling computations that remain out of
practical reach for the monolithic method.