Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Tensor Network
QuantumCircuitOperator[...]["TensorNetwork"] | returns the tensor network of a circuit |
TensorNetworkIndexGraph[...] | transforms a tensor network into a new graph with indices as vertices |
TensorNetworkFreeIndices[...] | returns free indices in a tensor network index graph |
ContractTensorNetwork[...] | contracts indices in a tensor network |
Create a quantum circuit:
In[58]:=
circuit=
[{"S","H"2,"X"3,"CNOT","SWAP"{2,3},{1},{2},{3}}];circuit["Diagram"]
 | QuantumCircuitOperator |
Out[59]=
Note in Wolfram quantum framework, each measurement result is saved in an ancillary quantum system (call it a detector, for example). So the true diagram will be something like this:
circuit["Diagram","ShowExtraQudits"True]
Note the first measurement is saved in the wire labeled as 0 and the rest goes backward, with negative labels.
Return the tensor network representation of the circuit:
net=circuit["TensorNetwork",GraphLayout{"LayeredDigraphEmbedding","Orientation"Left}]
The tensor network is a graph annotated with tensors and contraction indices.
GraphQ[net]&&TensorNetworkQ[net]
True
Lists all annotation keys available for the tensor network:
AnnotationKeys[{net,0}]
{Index,Tensor,VertexCoordinates,VertexLabels,VertexShape,VertexShapeFunction,VertexSize,VertexStyle}
Let’s look at tensor, vertex labels, and corresponding indexes in the tensor network:
list=Developer`FromPackedArray@VertexList[net];TableFormTranspose[Prepend[AnnotationValue[{net,list},#]&/@{"Tensor","Index"},Sort@AnnotationValue[net,VertexLabels]]],
VertexLabel | Tensor | Index | |||||
-20 | SparseArray
| { 3 -2 | |||||
-10 | SparseArray
| { 2 -1 | |||||
00 | SparseArray
| { 1 0 | |||||
1S | SparseArray
| { 1 1 1 1 | |||||
2H | SparseArray
| { 2 2 2 2 | |||||
3X | SparseArray
| 3 3 3 3 | |||||
4C⊗NOT | SparseArray
| { 1 4 2 4 4 1 4 2 | |||||
5SWAP | SparseArray
| 2 5 3 5 5 2 5 3 | |||||
6 Measurement 1 | SparseArray
| { 0 6 1 6 6 1 | |||||
7 Measurement 2 | SparseArray
| { -1 7 2 7 7 2 | |||||
8 Measurement 3 | SparseArray
| -2 8 3 8 8 3 |
Note tensors in the tensor network are mixed type, meaning they consist of so-called “contravariant” (upper) indices and “covariant” (lower) indices. For example, the 2nd measurement (7th vertex), acts on qubit-2 (denoted by contravariant and covariant indices 2) and its result is saved on wire, denoted by the index “-1”.
Vertices correspond to circuit’s operators/gate indices, in addition to “Initial” tensor with index 0 (for the initial state):
VertexList[net]
{-2,-1,0,1,2,3,4,5,6,7,8}
Length@circuit["Flatten"]["Operators"]
8
%Max[%%]
True
Note that each edge represents (i.e., is tagged by) a contraction:
EdgeList[net]
01,-12,-23,14,24,45,35,46,57,58
,
1
0
1
1
,
2
-1
2
2
,
3
-2
3
3
,
1
1
4
1
,
2
2
4
2
,
2
4
5
2
,
3
3
5
3
,
1
4
6
1
,
2
5
7
2
,
3
5
8
3
Perform the contraction:
finalTensor=ContractTensorNetwork[net]
SparseArray
 |  |
|
Confirm that the result is the same as default circuit application:
circuit[]["Tensor"]finalTensor
True
Another tensor network representation uses indices as graph vertices with tensors as cliques:
indexNet=TensorNetworkIndexGraph[net,GraphLayout{"LayeredDigraphEmbedding","Orientation"Left}]
Note in above graph, the directed edges imply tensor contraction; also tensors are cliques in above graph
Free indices are the ones that left after contraction:
Free indices can be extracted as vertices with zero in- and out- degree:
Contraction and Einstein Summation
Contraction and Einstein Summation
What ContractTensorNetwork does is in fact EinsteinSummation.
Show ContractTensorNetwork is the same as EinsteinSummation:
One can compare the performance, on how the relevant computation is done
Perform the contraction in the order of network’s EdgeList:
Optimize the order for contraction, using EinsteinSummation and symbolic tensors package:
Initial state different from ground state
Initial state different from ground state
Note that the initial tensor in the tensor network we studied here was a registered state. Additionally, one can start from any initial state.
Generate a random state:
Initialize the tensor network from above state:
See supplement info, for package-scoped symbols.
Show the tensor contraction is the same as transformation of state by the circuit:
Supplement info
Package-scoped symbols
Package-scoped symbols
FromTensorNetwork
FromTensorNetwork
Any directed graph can be turned into a tensor network, even if the graph is not annotated.