Parameter Trajectories for Quantum Natural Gradient Descent

Compare quantum natural gradient descent and regular gradient descent

In the following example we will illustrate how to apply the quantum natural gradient descent optimization for a parametrized single qubit state and visualize how it overperforms the regular gradient descent algorithm.

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:

=Re

∂



∂

-

∂







∂



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 the following ansatz for a single qubit state:

In[4]:=

SingleQubit=QuantumState[{Cos[θ1],Exp[2*I*θ2]*Sin[θ1]},"Parameters"{θ1,θ2}];SingleQubit["Formula"]

Out[4]=

Cos[θ1]|0〉+

2θ2



Sin[θ1]|1〉

Implement the

|ϕ(θ)〉

state:

In[5]:=

〉

=SingleQubit["StateVector"];

In[6]:=

〈

=SingleQubit["Dagger"]["StateVector"];

Calculate the Fubini-Study metric tensor:

In[7]:=

FubinyMetricTensor[{θ1_,θ2_}]=Table[ComplexExpand[Re[D[

〈

,θi].D[

〉

,θj]]-D[

〈

,θi].

〉

〈

.D[

〉

,θj]],{θi,{θ1,θ2}},{θj,{θ1,θ2}}]//Normal;

In[8]:=

Simplify@FubinyMetricTensor[{θ1,θ2}]//MatrixForm

Out[8]//MatrixForm=

1	0
0	2 Sin[2θ1]

Implement the cost function considering

In[9]:=

CostFunction[θ1_,θ2_]=

〈

.PauliMatrix[1].

〉

;

In[10]:=

FullSimplify[CostFunction[θ1,θ2],{θ1,θ2}∈Reals]

Out[10]=

Cos[2θ2]Sin[2θ1]

Obtain the correspondent gradient for each

In[11]:=

CostFunctionGradient[θ1_,θ2_]=Grad[FullSimplify[CostFunction[θ1,θ2],{θ1,θ2}∈Reals],{θ1,θ2}]

Out[11]=

{2Cos[2θ1]Cos[2θ2],-2Sin[2θ1]Sin[2θ2]}

Calculate the quantum and regular gradient descent for initial parameters



:

In[12]:=

QuantumParameters=QuantumNaturalGradientDescent[CostFunction,{π/12,π/12},FubinyMetricTensor,"Jacobian"CostFunctionGradient,"LearningRate"0.05,"MaxIterations"50];

In[13]:=

RegularParameters=

GradientDescent[

CostFunction,{π/12,π/12},

"Jacobian"CostFunctionGradient,

"LearningRate"0.05,

"MaxIterations"50]

;

The visualization shows that

is a singularity. The quantum natural gradient descent finds the point



={π/4,π/2}

as the optimum point closest to

. This does not cross the singular point, in contrast with the regular gradient descent:

In[14]:=

ListLinePlot{QuantumParameters,RegularParameters},



Out[14]=

The ground state is reached in fewer steps by the quantum gradient descent:

In[15]:=

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



Out[15]=

Calculate the quantum and regular gradient descent for initial parameters



5π

:

In[16]:=

QuantumParameters=QuantumNaturalGradientDescent[CostFunction,{5π/12,π/12},FubinyMetricTensor,"Jacobian"CostFunctionGradient,"LearningRate"0.05,"MaxIterations"50];

In[17]:=

RegularParameters=

GradientDescent[

CostFunction,{5π/12,π/12},

"Jacobian"CostFunctionGradient,

"LearningRate"0.05,

"MaxIterations"50]

;

According to the visualization, now the singularity is found at

=π/2

. The quantum gradient descent result is the same as the last example, the regular algorithm fails to notice the singularity:

In[18]:=

ListLinePlot{QuantumParameters,RegularParameters},



Out[18]=

Again, the ground state is reached in fewer steps by the quantum gradient descent:

In[19]:=

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



Out[19]=

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

Parameter Trajectories for Quantum Natural Gradient Descent

External Links

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