Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`
MPSCanonicalQ  |
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Examples
(3)
Scope
(1)
Build one MPS and canonicalize it three ways:
In[1]:=
mps2=BlockRandomSeedRandom[42];
["MPS"[6,4,2]]
 | RandomTensorNetwork |
In[2]:=
 | MPSCanonicalQ |
 | MPSCanonicalForm |
Out[2]=
True
In[3]:=
| MPSCanonicalQ |
| MPSCanonicalForm |
Out[3]=
True
In[4]:=
| MPSCanonicalQ |
| MPSCanonicalForm |
Out[4]=
True
A raw uncanonicalized MPS is not in any of the three forms:
In[1]:=
mps2=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];
[mps2,#]&/@{"Left","Right",{"Mixed",3}}
 | RandomTensorNetwork |
| MPSCanonicalQ |
Out[1]=
{False,False,False}
Perturb a left-canonical MPS by a small amount; with the default tolerance the perturbed network no longer passes:
In[2]:=
mps2=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];perturbed=BlockRandomSeedRandom[7];left=
[mps2,"Left"];
[Map[#+RandomReal[{-1,1},Dimensions[#]]&,left["Tensors"]],left["Hyperedges"],left["Output"]]
| RandomTensorNetwork |
 | MPSCanonicalForm |
| TensorNetwork |
-8
10.
In[3]:=
| MPSCanonicalQ |
Out[3]=
False
Relaxing the tolerance to absorbs the noise:
-6
10
In[1]:=
mps2=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];perturbed=BlockRandomSeedRandom[7];left=
[mps2,"Left"];
[Map[#+RandomReal[{-1,1},Dimensions[#]]&,left["Tensors"]],left["Hyperedges"],left["Output"]];
[perturbed,"Left",]
| RandomTensorNetwork |
| MPSCanonicalForm |
| TensorNetwork |
-8
10.
| MPSCanonicalQ |
-6
10.
Out[1]=
True
In[2]:=
mps2=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];
[mps2,{"Mixed",3}],{"Mixed",100}
| RandomTensorNetwork |
| MPSCanonicalQ |
| MPSCanonicalForm |
Out[2]=
False
Centers at the boundaries and are valid; they reduce to pure right- and left-canonical forms:
k=1
k=n
In[1]:=
mps2=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];
[mps2,"Right"],{"Mixed",1}
| RandomTensorNetwork |
| MPSCanonicalQ |
| MPSCanonicalForm |
Out[1]=
True
A left-canonical MPS typically fails the right-isometry check, since the two forms are related by a non-trivial gauge transformation:
In[2]:=
mps2=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];
[mps2,"Left"],"Right"
| RandomTensorNetwork |
| MPSCanonicalQ |
| MPSCanonicalForm |
Out[2]=
False
Applications
(1)
Properties & Relations
(1)
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Start from a six-site MPS with bond dimension 4 and physical dimension 2:
In[1]:=
mps=BlockRandomSeedRandom[42];
["MPS"[6,4,2]]
| RandomTensorNetwork |
The left-canonical form passes the left-isometry check:
In[2]:=
leftMPS=
[mps,"Left"]
 | MPSCanonicalForm |
In[3]:=
| MPSCanonicalQ |
Out[3]=
True
The right-canonical form passes the right-isometry check:
In[4]:=
mps=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];rightMPS=
[mps,"Right"]
| RandomTensorNetwork |
| MPSCanonicalForm |
In[5]:=
| MPSCanonicalQ |
Out[5]=
True
Mixed-canonical form at site places the orthogonality center there; sites 1 and 2 are left-isometric, sites 4-6 are right-isometric:
k=3
In[6]:=
mps=BlockRandomSeedRandom[42];
["MPS"[6,4,2]];mixedMPS=
[mps,{"Mixed",3}]
| RandomTensorNetwork |
| MPSCanonicalForm |
In[7]:=
| MPSCanonicalQ |
Out[7]=
True
""