Quantum Natural Gradient Descent via Variational Quantum Eigensolver

Optimize a variational quantum circuit using quantum natural gradient descent through VQE

In the following example we will illustrate how to apply the quantum natural gradient descent optimization for parametrized variational quantum circuits by calculating the Fubini–Study metric tensor through the variational quantum eigensolver approach.

The quantum state space features a distinct invariant metric tensor referred to as the Fubini-Study metric tensor

, which can be used to develop a quantum version of natural gradient descent. Then each optimization step is given by:

t+1

-η

(θ)∇ℒ(θ)

where ℒ is the cost function of parameter θ,

the Fubini-Study metric tensor pseudo-inverse and η the step learning size.

The fundamental process of the Variational Quantum Eigensolver (VQE) involves using a short quantum circuit U(θ), characterized by parameters θ =

{

,…,

}

, to iteratively adjust θ to minimize the average energy or cost function:

f(θ) =

〈ϕ(θ)|H|ϕ(θ)〉

where

|ϕ(θ)〉=U(θ)|0〉.

In the VQE setting the Fubini-Study metric follows:

=Re

∂



∂

-

∂







∂



where

∂

ϕ(θ)〉=

∂

|ϕ(θ)〉/∂

Install the QuantumFramework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"]

Load the paclet:

In[2]:=

Needs["Wolfram`QuantumFramework`"]

Load additional functions used for examples, explained in this Tech Note:

In[3]:=

Needs["Wolfram`QuantumFramework`ExampleRepository`"]

Consider a simple variational circuit that consists of 4 parameters with 2 parametrized layers:

In[4]:=

Implement the variational quantum circuit:

In[5]:=

initParameters={0.432,-0.123,0.543,0.233};

In[6]:=

VQC=QuantumCircuitOperator "RY"

1,"RY"

2,"RY"

3, "RZ"[θ1]1,"RZ"[θ2]2,"CNOT"{1,2},"CNOT"{2,3}, "RY"[θ3]2,"RX"[θ4]3,"CNOT"{1,2},"CNOT"{2,3},"Parameters"{θ1,θ2,θ3,θ4};

Calculate the Fubini-Study metric tensor through the Variational Quantum Eigensolver method using the FubiniStudyMetricTensor function:

In[7]:=

VQCState=VQC[];

In[8]:=

VQCFubini[{θ1_,θ2_,θ3_,θ4_}?(VectorQ[#,NumericQ]&)]=FubiniStudyMetricTensor[VQCState,{θ1,θ2,θ3,θ4}];

Test using the initial parameters:

In[9]:=

Chop[VQCFubini[initParameters],

-10

]//Re//MatrixForm

Out[9]//MatrixForm=

0.125	0	-0.00576267	0
0	0.1875	0.00407482	0
-0.00576267	0.00407482	0.249734	-0.015247
0	0	-0.015247	0.202936

Implement the cost function with

11

In[10]:=

VQCKet=VQCState["StateVector"];

In[11]:=

VQCBra=VQCState["Dagger"]["StateVector"];

In[12]:=

CostFunction[θ1_,θ2_,θ3_,θ4_]=VQCBra.QuantumOperator[{"PauliY"{1},"I"{2},"I"{3}}]["Matrix"].VQCKet;

In[13]:=

CostFunction[θ1,θ2,θ3,θ4]//ComplexExpand//FullSimplify

Out[13]=

Cos[θ3]Sin

Sin[θ1]+

Cos[θ1]Sin[θ2]Sin[θ3]

Calculate the correspondent gradient for each

In[14]:=

CostFunctionGrad[θ1_,θ2_,θ3_,θ4_]=Grad[CostFunction[θ1,θ2,θ3,θ4]//ComplexExpand//Simplify,{θ1,θ2,θ3,θ4}]

Out[14]=



2Cos[θ1]

Cos

θ3



Sin

-2Cos[θ1]Sin



Sin

θ3



Sin[θ1]Sin[θ2]Sin[θ3],

Cos[θ1]Cos[θ2]Sin[θ3],

Cos[θ1]Cos[θ3]Sin[θ2]-4Cos

θ3

Sin

Sin[θ1]Sin

θ3



,0

Compute the quantum natural gradient descent using the QuantumNaturalGradientDescent function:

In[15]:=

QuantumParameters=QuantumNaturalGradientDescent[CostFunction,initParameters,VQCFubini,"Jacobian"CostFunctionGrad,"LearningRate"0.01,"MaxIterations"200];//AbsoluteTiming

Out[15]=

{4.67015,Null}

Compute a regular gradient descent using the GradientDescent function:

In[16]:=

RegularParameters=

GradientDescent[CostFunction,initParameters,

"Jacobian"CostFunctionGrad,

"LearningRate"0.01,

"MaxIterations"200]

;

Visualize and compare the quantum and regular optimization using the parameters obtained in the previous calculation:

In[17]:=

ListLinePlot{CostFunction@@@QuantumParameters,CostFunction@@@RegularParameters},



Out[17]=

External Links

Source Metadata

Citation:
- Natural gradient descent optimization for quantum circuits with Fubiny–Study metric tensor
  by Sebastian Rodriguez
  Wolfram Community, STAFF PICKS, February 28, 2024 https://community.wolfram.com/groups/-/m/t/3131997
- Yamamoto, N. (2019). On the natural gradient for variational quantum eigensolver (Version 1). arXiv. https://doi.org/10.48550/ARXIV.1909.05074
- Izaac, J. (2019, October 10). Quantum natural gradient. PennyLane Demos. https://pennylane.ai/qml/demos/tutorial_quantum_natural_gradient/

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Contributed by: Wolfram Research, Quantum Computation Framework Team

Wolfram Language Example Repository

Quantum Natural Gradient Descent via Variational Quantum Eigensolver

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