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Quantum Computation

Hermitian into Unitary Transformation

Use the parameter-shift rule to implement a unitary gate corresponding to an operator

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Utilizing the parameter-shift rule involves transforming the eigenvalues of a Hermitian operator into ±1, thereby converting it into a unitary operator. This technique is widely employed for measuring gradients of quantum circuits in relation to their parameters.

Install and load the QuantumFramework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"];Needs["Wolfram`QuantumFramework`"];

Generate a pseudorandom GUE matrix:

In[2]:=

h=RandomVariate[GaussianUnitaryMatrixDistribution[2]]

Out[2]=

{{0.257008+0.,0.158164+0.26653},{0.158164-0.26653,1.35714+0.}}

Check if it is unitary or Hermitian:

In[3]:=

Comap[{UnitaryMatrixQ,HermitianMatrixQ},h]

Out[3]=

{False,True}

Calculate its eigenvalues:

In[4]:=

{e0,e1}=Eigenvalues[h]

Out[4]=

{1.43845,0.175705}

Transform the matrix as follows, by shifting the eigenvalues:

In[5]:=

h2=

e0-e1

(h-1/2(e1+e0)IdentityMatrix[2])

Out[5]=

{{-0.871228+0.,0.250509+0.422145},{0.250509-0.422145,0.871228+0.}}

Check if it is unitary or Hermitian:

In[6]:=

Comap[{UnitaryMatrixQ,HermitianMatrixQ},h2]

Out[6]=

{True,True}

A similar calculation can be done in quantum framework by changing the basis. Define a basis as eigenvectors of the Hermitian operator:

In[7]:=

QuantumBasis[Eigenvectors@h]

Out[7]=

QuantumBasis

Picture: Schrödinger	Rank: 1
Dimension: 2



Create a diagonalize operator, with eigenvalues ±1 in the new basis:

In[8]:=

h3=QuantumOperator[PauliMatrix[3],QuantumBasis[Eigenvectors@h]]

Out[8]=

QuantumOperator

Picture: Schrödinger	Arity: 1
Dimension: 2→2	Qudits: 1→1
Hermitian: True	Order: {1}→{1}
Unitary: True	Dimensions: {2}→{2}
ParameterArity: 0	Parameters: {}



As one can see in the summary box, the new operator is both Hermitian and unitary; and its matrix in the computational basis is the same as h2:

In[9]:=

h3["Computational"]["Matrix"]-h2//Chop

Out[9]=

{{0,0},{0,0}}

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Contributed by: Wolfram Research, Quantum Computation Framework Team

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Hermitian into Unitary Transformation

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