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Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
WickUnitary  |  |
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Details and Options
Examples
(5)
Basic Examples
(1)
Consider some fermion modes.
In[1]:=
$n=3;
Consider a unitary operator.
In[2]:=
SeedRandom[357];
In[3]:=
ham=Re@RandomAntisymmetric[2*$n];wu=WickUnitary[MatrixExp[ham]]
Out[3]=
WickUnitary
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Get the matrix representation in the computational basis.
In[4]:=
mat=Matrix[wu]//Chop;mat//ArrayShort
Out[4]//MatrixForm=
0.761713+0.479101 | 0 | 0 | -0.0958183-0.146774 | … |
0 | 0.444503+0.37586 | -0.691974+0.314491 | 0 | … |
0 | 0.170742+0.614613 | 0.462074+0.256298 | 0 | … |
0.292814+0.130956 | 0 | 0 | 0.744422+0.324297 | … |
… | … | … | … | … |
Pick an initial state.
In[5]:=
in=RandomWickState[$n]
Out[5]=
WickState
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Operate the unitary operator on the initial state.
In[6]:=
out=wu**in
Out[6]=
WickState
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Convert the results to the vector representation.
In[7]:=
vin=Matrix[in];vin//ArrayShort
Out[7]=
{0,0.597393,0.33844+0.2461,0,…}
In[8]:=
vout=Matrix[out];vout//ArrayShort
Out[8]=
{0,0.183398,0.470325+0.0187807,0,…}
In[9]:=
mint=mat.vin//Chop;mint//ArrayShort
Out[9]=
{0,-0.00654079+0.183282,-0.0355426+0.469356,0,…}
The two results vout and mint look different, but that is due to a global phase factor. Note that the matrix representation of is always in the canonical form.
In[10]:=
vout-CanonicalizeVector[mint]//ArrayZeroQ
Out[10]=
True
Now, let us verify the above result using the conventional method.
Consider a specific set of fermion modes for analysis.
In[11]:=
Let[Fermion,a]aa=a[Range@$n]
Out[11]=
{,,}
a
1
a
2
a
3
In[12]:=
Let[Majorana,c]cc=c@Range[2$n]
Out[12]=
{,,,,,}
c
1
c
2
c
3
c
4
c
5
c
6
In[13]:=
hamOp=MultiplyDot[cc,ham.cc]*I/4;hamOp=ToDirac[hamOp,ccaa];uni=MatrixExp[-I*Matrix[hamOp,aa]]
Out[13]=
SparseArray
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In[14]:=
uni-mat//Chop//ArrayZeroQ
Out[14]=
True
Scope
(4)
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""