Concept record · edition 0.1.0
Tensor network
A representation of a high-rank tensor as a contracted network of lower-rank tensors, whose graph structure encodes which degrees of freedom are directly correlated.
Why it matters
It replaces a state vector that grows exponentially in system size with a set of small tensors whose size is controlled by a chosen rank parameter.
What this does not establish
The representation is efficient only for states whose correlation structure matches the network geometry. It is not a general-purpose compression of arbitrary quantum states.
Claims using this concept
- tn-001 · Established resultA matrix product state represents a many-body state with a cost controlled by its bond dimension, which bounds the entanglement it can carry across any cut.
- tn-008 · Established resultThe cost of contracting a tensor network is governed by the contraction order and by structural properties of the network graph such as its treewidth.
Sources
review article · identifier verified
A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
Román Orús · 2013
arXiv:1306.2164
Review covering tensor-network structure, entanglement scaling, and contraction strategy; the general reference for this atlas.
Source record →review article · identifier verified
Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
F. Verstraete, J. I. Cirac, V. Murg · 2008
arXiv:0907.2796
Review tying MPS, PEPS, and variational renormalization group methods into one framework.
Source record →