2. Contraction¶

one of the core tasks of any tensor network algorithm: collection of tensors -> single tensor

defined as sum of products $$ \sum_{\sigma} \prod_{i} T_i(\sigma_i) $$

exponentially slow to do sum directly since we would have evaluate every single combination of index value assignments.

Instead pairwise path of intermediates always best cost wise -> contracting a pair of tenosrs can remove indices entirely from the rest of the contraction

specified by “contraction tree”, cost is incredibly sensitive to the choice and the space of these trees is very large -> tricky problem, but can be automated.

Tradeoff between time spent finding the path and the time spent actually doing the contraction.

The optimize kwarg.

2.1. contraction interfaces¶

qtn.tensor_contract
ta @ tb for simple two tensor contract
TensorNewtork.contract

special forms of contraction

And information methods:

TensorNetwork.contraction_path
the opt_einsum path
TensorNetwork.contraction_info
the opt_einsum.PathInfo object
TensorNetwork.contraction_tree
the cotengra.ContractionTree object
TensorNetwork.contraction_width
log2 of the maximum size of any intermediate tensor.
TensorNetwork.contraction_cost
the total number of scalar operations required to perform the contraction

called within many algorithms (wherever you see the optimize kwarg)

2.2. Things you can supply to the `optimize` kwarg:¶

2.2.1. `str` preset¶

2.2.2. `opt_einsum` `PathOptimizer`¶

2.2.3. `cotengra` `HyperOptimizer`¶

2.2.4. explicit contraction path¶

2.2.5. path caching¶

geometry hash

caching within cotengra

2.3. Hyper edges¶

most general einsum equation

2.4. Structured Contractions¶

e.g. 1D chain

2.5. Approximate boundary contraction¶

coarse graining contractions

hyper edges not supported

2.6. Automatic approximate / compressed contraction¶

contract_copmressed
contract_around