Quantum Circuit Noise Parameter Estimation

Estimate the parameters of the quantum channels modeling a noisy environment with optimization functions

Given an experimental value (target) and information about the type of noise channel that affects each qubit, it is possible to estimate the values of the channel parameters that correspond to the measured expected value.

Install the Quantum Framework Paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"]

Load the paclet and the functions explained in this Tech Note:

In[2]:=

Get["Wolfram`QuantumFramework`"]Needs["Wolfram`QuantumFramework`ExampleRepository`"]

Build a circuit that simulates the environment. In this example there are two qubits with AmplitudeDamping and BitFlip channels. Include the noise parameters as an option to QuantumCircuitOperator:

In[3]:=

qc=QuantumCircuitOperator[{QuantumCircuitOperator[{"H"1,"CNOT"{1,2}}],QuantumCircuitOperator[{QuantumChannel[{"AmplitudeDamping",γ},{1}],QuantumChannel[{"BitFlip",p},{2}]}]},"Parameters"{γ,p}];

In[4]:=

qc["Diagram",ImageResizeSmall]

Out[4]=

Use the target value and evaluate the circuit. A cost function is also needed to know how far the expected value of



is from the target:

In[5]:=

target=0.6582;state=qc[];costFunc[x_,y_]:=((QuantumMeasurementOperator["Z"{1,-1,-1,1},{2}]@state[γx,py])["Mean"]-target)^2

Use the GradientDescent function with the cost function and an initial guess:

In[6]:=

gradResults=GradientDescent[costFunc,{0.4,0.35},"MaxIterations"25,"LearningRate"0.1,"Jacobian"None];

Get the channel parameters and compare with the standard numerical optimization function:

In[7]:=

{gradResults〚-1〛,params=NMinimize[costFunc[γ,p],0≤γ≤0.5&&0≤p≤0.5,{γ,p}]〚2〛}

Out[7]=

{{0.276331,0.0452342},{γ0.232013,p0.0714772}}

Verify the cost function is close to 0:

In[8]:=

{costFunc[Sequence@@gradResults〚-1〛],costFunc[Sequence@@({γ,p}/.params)]}

Out[8]=

{3.31448×

-19

,2.6058×

-20

}

Visualize the parameter convergence obtained by gradient descent:

In[9]:=

ListLinePlot{gradResults〚All,1〛,gradResults〚All,2〛},



Out[9]=

GradientDescent and

NMinimize

both find acceptable solutions to this problem, however, the GradientDescent function from the framework is more suitable to this problem.

External Links

Wolfram/QuantumFramework

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Contributed by: Bruno Tenorio, Wolfram Research, Quantum Computation Framework Team

Wolfram Language Example Repository

Quantum Circuit Noise Parameter Estimation

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Quantum Circuit Noise Parameter Estimation

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