Concept record · edition 0.1.0
Bond dimension (D)
The rank retained on each virtual index of a tensor network, bounding the entanglement the state can carry across a cut at log(D).
Why it matters
It is the single parameter trading accuracy against cost, and the number that must accompany any tensor-network result for it to be interpretable.
What this does not establish
A larger D is not a general fix. The D required to hold a fixed accuracy can grow exponentially with the entanglement of the target state.
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-002 · Established resultThe one-dimensional area law proves that ground states of gapped one-dimensional local Hamiltonians have entanglement bounded independently of system size.
- tn-004 · Established resultTensor-network accuracy is controlled by singular-value truncation, and the discarded weight is the quantity that makes a result interpretable.
- tn-010 · Established resultTensor-network time evolution is limited by entanglement growth, which forces the bond dimension upward as the simulated time increases.
Sources
review article · identifier verified
The density-matrix renormalization group in the age of matrix product states
Ulrich Schollwöck · 2010
arXiv:1008.3477
Standard reference for MPS canonical forms, SVD truncation, and the reformulation of DMRG in tensor-network language.
Source record →primary paper · identifier verified
Efficient classical simulation of slightly entangled quantum computations
Guifré Vidal · 2003
arXiv:quant-ph/0301063
Ties classical simulation cost to the entanglement carried across a cut, which is the quantity a bond dimension budgets.
Source record →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 →