Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`
TensorNetworkContraction  |
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▪
By default returns an expression tree () that you can inspect, transform, or pass to .
"Inactive"True
▪
Setting evaluates the tree and returns the numerical tensor; this is the path takes.
"Inactive"False
▪
ActivateTensors
▪
The option selects one of five binary-contraction engines: "ArrayDot" (default), "ArrayDotTranspose", "Dot", "TensorContract", and "TableSum". All engines produce numerically identical results after activation; they differ only in the internal representation of the tree. The full list is in .
$TensorNetworkContractionMethods
▪
For a single-tensor network (no contraction indices), returns the tensor directly via the fallback path.
▪
A is a list of integer pairs in opt_einsum convention; a is the tree-structured form produced by .
path
{{i,j},…}
treePath
▪
The following options can be given:
Examples
(1)
Build a fixed three-tensor network for the path-form examples:
In[2]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}]
 | TensorNetwork |
Out[3]=
TensorNetwork
 |  |
|
Default form auto-selects a path:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn]
 | TensorNetwork |
 | TensorNetworkContraction |
Out[2]=
TensorContract[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}}{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}}{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}},{{2,3},{4,5}}]
An explicit canonical path drives the binary contractions step by step:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,{{1,2},{1,2}}]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
Transpose[ArrayDot[{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}},ArrayDot[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{2,1}}],{{1,2}}],Cycles[{{1,2}}]]
Method-string shortcuts produce the path via GreedyContractionPath, OptimalContractionPath, or one of the cost variants:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,"Greedy"]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
ArrayDot[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},ArrayDot[{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}},{{2,1}}],{{2,1}}]
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,"flops"]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
ArrayDot[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},ArrayDot[{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}},{{2,1}}],{{2,1}}]
The five engines produce structurally distinct trees for the same path:
ArrayDot (default) emits the index-aware Inactive[ArrayDot] primitive:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,{{1,2},{1,2}},Method"ArrayDot"]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
Transpose[ArrayDot[{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}},ArrayDot[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{2,1}}],{{1,2}}],Cycles[{{1,2}}]]
ArrayDotTranspose introduces explicit Inactive[Transpose] reshuffles around each ArrayDot:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,{{1,2},{1,2}},Method"ArrayDotTranspose"]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
Transpose[ArrayDot[Transpose[{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}},Cycles[{{1,2}}]],Transpose[ArrayDot[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},1],Cycles[{{1,2}}]],1],Cycles[{{1,2}}]]
Dot reshapes operands to matrices and uses Inactive[Dot]:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,{{1,2},{1,2}},Method"Dot"]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
Transpose[Transpose[{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}},Cycles[{{1,2}}]].Transpose[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}}.{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},Cycles[{{1,2}}]],Cycles[{{1,2}}]]
TensorContract emits a TensorProduct wrapped in TensorContract:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,{{1,2},{1,2}},Method"TensorContract"]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
Transpose[TensorContract[{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}TensorContract[{{0.5,-0.2,0.3},{0.1,0.8,-0.4}}{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{2,3}}],{{1,4}}],Cycles[{{1,2}}]]
TableSum unrolls the contraction into explicit Table/Sum loops over formal indices:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];
[tn,{{1,2},{1,2}},Method"TableSum"]
| TensorNetwork |
| TensorNetworkContraction |
Out[2]=
TransposeTable{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}c.1,i.1Table{{0.5,-0.2,0.3},{0.1,0.8,-0.4}}i.1,c.1{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}}c.1,j.1,{i.1,2},j.1,4j.1,c.1,{i.1,2},j.1,2,Cycles[{{1,2}}]
4
∑
c.1
3
∑
c.1
All five engines yield the same numerical value after activation:
In[1]:=
tn=
[{{{0.5,-0.2,0.3},{0.1,0.8,-0.4}},{{0.6,0.4,-0.1,0.2},{-0.3,0.5,0.7,-0.2},{0.1,-0.4,0.3,0.5}},{{0.2,-0.3},{0.4,0.1},{-0.5,0.6},{0.3,-0.2}}},{{i,j},{j,k},{k,l}}];Equal@@
[tn,{{1,2},{1,2}},Method#1]&/@$TensorNetworkContractionMethods
| TensorNetwork |
 | ActivateTensors |
 | TensorNetworkContraction |
From a TensorNetwork object (the typical workflow):
From a tensor-network Graph:
From the low-level Association produced by TensorNetworkData on a binary network:
ContractionTree renders the contraction hierarchy of an inactive tree:
With "Labels" "Dimensions" the tree shows the running shape at each node:
Construct a small two-tensor network used in the examples below:
Return the default inactive contraction expression:
Evaluate the tree with ActivateTensors:
The option "Inactive" False evaluates the tree on the fly: