Adaptive Patching for Tensor Train Computations

TL;DR

This work proposes an adaptive patching scheme that exploits block-sparse QTT structures to reduce costs through divide-and-conquer, adaptively partitioning tensors into smaller patches with reduced bond dimensions.

Abstract

Quantics Tensor Train (QTT) operations such as matrix product operator contractions are prohibitively expensive for large bond dimensions. We propose an adaptive patching scheme that exploits block-sparse QTT structures to reduce costs through divide-and-conquer, adaptively partitioning tensors into smaller patches with reduced bond dimensions. We demonstrate substantial improvements for sharply localized functions and show efficient computation of bubble diagrams and Bethe-Salpeter equations, opening the door to practical large-scale QTT-based computations previously beyond reach.

Figures (21)

• Figure 1: MPO--MPO contraction. Two MPOs are contracted by contracting their external legs, and the resulting MPO is compressed on the fly. Here, $\mathcal{L}$ is the length of the MPOs.
• Figure 2: (a) Partitioning of the domain of the bivariate function $f(\boldsymbol{r})$, Eq. (\ref{['eq:localFunc']}), as obtained from the adaptive patching algorithm. We used the fused representation with local dimension $d\!=\!4$. Smaller maximum bond dimensions per patch $\chi_{\text{p}}$ yield finer partitionings of the domain. (b) Hierarchical tree that records the patch refinement: each tensor slice is attached to its parent, and red leaves mark the yet-to-converge patches produced by the intermediate steps of the refinement. (c) Number of total patches vs. bond-dimension cap $\chi_{\text{p}}$. The scaling $N_{\text{p}} \sim \chi_{\text{p}}^{-2}$ indicates that the number of total parameters is approximately constant for different choices of $\chi_{\text{p}}$, for both fused and interleaved representations (see the main text for more details).
• Figure 3: Patched QTCI approximation of $\operatorname{Re}G({\boldsymbol{k}})$ defined in Eq. \ref{['eq:2DGreen']}. (a) Heatmaps of the patched tensor train evaluated on $[-\pi,\pi]^{2}$ for bond-caps $\chi_{\text{p}}=48,118,277$ (corresponding to $\delta=10^{-1},10^{-2},10^{-3}$, respectively) at $\tau=10^{-7}$. (b) Pointwise error $\text{err}({\boldsymbol{k}})$ [Eq. \ref{['eq:localError2DGreen']}] for the same patched approximations, plotted using a $\log_{10}$ color scale. (c) Comparison of the bond-dimension profiles for the pQTCI (one line per patch) and standard QTCI representations of $\operatorname{Re}G({\boldsymbol{k}})$. Every patch in the pQTCI result adheres to the prescribed bond-cap $\chi_{\text{p}}$.
• Figure 4: Total parameter count (left) and CPU run time (right) versus bond-cap $\chi_{\text{p}}$ of a patched QTCI approximation of $\text{Re}\left(G({\boldsymbol{k}})\right)$ for $\delta=10^{-3}$, using interleaved (a) or fused (b) QTT index ordering. Dotted lines show the corresponding standard-QTCI measurements.
• Figure 5: Parameter and run-time ratios between the best patched approximation (at $\chi_{\text{p}}^{\mathrm{best}}$) and a single-TT QTCI approximation, as a function of the broadening $\delta$.