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Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`
MPSCanonicalQ  |
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Details and Options
Examples
(3)
Scope
(1)
Build one MPS and canonicalize it three ways:
In[1]:=
mps2=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]]
In[2]:=
MPSCanonicalQ[MPSCanonicalForm[mps2,"Left"],"Left"]
Out[2]=
True
In[3]:=
MPSCanonicalQ[MPSCanonicalForm[mps2,"Right"],"Right"]
Out[3]=
True
In[4]:=
MPSCanonicalQ[MPSCanonicalForm[mps2,{"Mixed",3}],{"Mixed",3}]
Out[4]=
True
A raw uncanonicalized MPS is not in any of the three forms:
In[1]:=
mps2=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];MPSCanonicalQ[mps2,#]&/@{"Left","Right",{"Mixed",3}}
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=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];perturbed=BlockRandom[SeedRandom[7];(left=MPSCanonicalForm[mps2,"Left"];TensorNetwork[Map[#+RandomReal[{-1,1},Dimensions[#]]&,left["Tensors"]],left["Hyperedges"],left["Output"]])]
-8
10.
In[3]:=
MPSCanonicalQ[perturbed,"Left"]
Out[3]=
False
Relaxing the tolerance to absorbs the noise:
-6
10
In[1]:=
mps2=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];perturbed=BlockRandom[SeedRandom[7];(left=MPSCanonicalForm[mps2,"Left"];TensorNetwork[Map[#+RandomReal[{-1,1},Dimensions[#]]&,left["Tensors"]],left["Hyperedges"],left["Output"]])];MPSCanonicalQ[perturbed,"Left",]
-8
10.
-6
10.
Out[1]=
True
In[2]:=
mps2=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];MPSCanonicalQ[MPSCanonicalForm[mps2,{"Mixed",3}],{"Mixed",100}]
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=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];MPSCanonicalQ[MPSCanonicalForm[mps2,"Right"],{"Mixed",1}]
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=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];MPSCanonicalQ[MPSCanonicalForm[mps2,"Left"],"Right"]
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=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]]
The left-canonical form passes the left-isometry check:
In[2]:=
leftMPS=MPSCanonicalForm[mps,"Left"]
In[3]:=
MPSCanonicalQ[leftMPS,"Left"]
Out[3]=
True
The right-canonical form passes the right-isometry check:
In[4]:=
mps=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];rightMPS=MPSCanonicalForm[mps,"Right"]
In[5]:=
MPSCanonicalQ[rightMPS,"Right"]
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=BlockRandom[SeedRandom[42];RandomTensorNetwork["MPS"[6,4,2]]];mixedMPS=MPSCanonicalForm[mps,{"Mixed",3}]
In[7]:=
MPSCanonicalQ[mixedMPS,{"Mixed",3}]
Out[7]=
True
""