Concepts

This page introduces the core ideas behind tensor4all-rs in plain language. No deep mathematical background is assumed.

Tensor Train (TT / MPS)

A tensor train (TT) represents a high-dimensional tensor as a chain of smaller, 3-index tensors. Instead of storing all entries of a size-d^N tensor (which grows exponentially with N), a TT stores N tensors each of modest size, multiplied together to reproduce any entry on demand. The size of the shared “bond” indices between adjacent tensors is called the bond dimension m; larger m means a more accurate approximation at the cost of more memory and compute. In quantum physics the same structure is called a Matrix Product State (MPS).

A[0] ---- A[1] ---- A[2] ---- ... ---- A[N-1]
  |          |          |                  |
 i_0        i_1        i_2             i_{N-1}

Vertical lines are site indices (also called physical indices) — one per tensor, labeling the dimension of the original tensor at that position. Horizontal lines are link indices (bond indices) — internal indices that connect neighboring tensors and whose size is the bond dimension.

Tensor Cross Interpolation (TCI)

Tensor Cross Interpolation approximates a high-dimensional function f(i_0, i_1, ..., i_{N-1}) by evaluating it at an adaptively chosen subset of points and fitting a tensor train. The idea generalises the CUR matrix decomposition — which approximates a matrix using selected rows and columns — to arbitrary numbers of dimensions. The algorithm automatically identifies the most informative “pivots” (evaluation points), so it never needs to query the function at every grid point. For smooth or structured functions, the number of required evaluations can be orders of magnitude smaller than the full grid. The output is a tensor train whose accuracy is controlled by a relative tolerance threshold (rtol).

Quantics Tensor Train (QTT)

Quantics (also called quantized tensor train) is a technique for representing functions of continuous variables with a tensor train. A function f(x) sampled on a uniform grid of 2^R points in [0, 1) is reshaped into an R-site tensor train where every site index has dimension 2. Each site corresponds to one bit of the binary representation of the grid index. Smooth functions have low bond dimension in this representation, giving exponential compression relative to storing all 2^R values. Combining QTT with TCI (“Quantics TCI”) allows efficient approximation of high-dimensional continuous integrands without ever forming the full grid.

bit R-1        bit R-2                bit 0
  |              |          ...          |
 B[0] --------- B[1] ----- ... ------- B[R-1]

Each tensor B[k] has two physical legs of dimension 2 (one per variable dimension in the multivariate case) and one or two bond legs.

Tree Tensor Network (TreeTN)

A Tree Tensor Network generalises the tensor train to tree-shaped graphs. Each node of the tree holds a tensor, and each edge corresponds to a shared (bond) index between the two tensors at its endpoints. Contracting along any edge multiplies those two tensors and merges their free indices. The tensor train is the special case where the tree is a simple path graph. Tree structures can capture correlations that are not naturally captured by a chain, making them useful for problems with hierarchical or multi-scale structure. In tensor4all-rs, TreeTensorNetwork<V> (where V is a vertex label type) is the primary data structure representing both TT/MPS and more general tree networks.

         T[root]
        /        \
    T[a]          T[b]
    /   \            \
T[c]   T[d]         T[e]
  |      |             |
 i_c    i_d           i_e

Vertical lines are site indices; lines along the tree edges are bond indices.

Key Terminology