Quantum Natural Gradient Descent via Block Diagonalization

Optimize a variational quantum circuit using quantum natural gradient descent through block diagonalization

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.

We can calculate

using an approximation of the Fubini-Study metric tensor for a variational quantum circuit, named the block-diagonal approach:

(l)

=〈

l-1

〉-〈

l-1

〉〈

l-1

〉

The circuit is given by:

U(θ)|

〉=

(

)

…

(

)

…

(

)

〉

where W are layers of non-parametrized gates and

(l)

are the generators of the parametrized gates of the form

(l)

θ

(l)



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]:=

QuantumCircuitOperator[{QuantumCircuitOperator[{QuantumState["0"],QuantumState["0"]{2},QuantumState["0"]{3},"RY"[π/4]1,"RY"[π/3]2,"RY"[π/7]3},"Label""Initial State Preparation \!\(\*SubscriptBox[\(W\), \(0\)]\)"],"Barrier",QuantumCircuitOperator[{"RZ"[θ1],"RZ"[θ2]2,"I"3},"Label""Parametrized layer 1"],QuantumCircuitOperator[{"CNOT"{1,2},"CNOT"{2,3}},"Label""\!\(\*SubscriptBox[\(W\), \(1\)]\)"],"Barrier",QuantumCircuitOperator[{"I"1,"RY"[θ3]2,"RX"[θ4]3},"Label""Parametrized layer 2"],QuantumCircuitOperator[{"CNOT"{1,2},"CNOT"{2,3}},"Label""\!\(\*SubscriptBox[\(W\), \(2\)]\)"],QuantumMeasurementOperator["Y",{1}]},"Parameters"{θ1,θ2,θ3,θ4}]["Diagram",ImageSizeLarge]

Out[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};

The Fubini-Study metric tensor

according to the block-diagonal approach:

(1) g	0
0	(2) g

First layer circuit diagram:

In[7]:=

QuantumCircuitOperator[{QuantumCircuitOperator[{"RY"[π/4]1,"RY"[π/3]2,"RY"[π/7]3},"Label""\!\(\*SubscriptBox[\(W\), \(0\)]\)"],QuantumMeasurementOperator["PauliZ",{1}],QuantumMeasurementOperator["PauliZ",{2}]}]["Diagram"]

Out[7]=

Calculate

(1)

using the following formula:

(1)

〈 2 Z 1 〉- 2 〈 Z 1 〉	〈 Z 1 Z 2 〉-〈 Z 1 〉〈 Z 2 〉
〈 Z 2 Z 1 〉-〈 Z 2 〉〈 Z 1 〉	〈 2 Z 2 〉- 2 〈 Z 2 〉

Implement functions to calculate the variance



–

〈A〉

and covariance

〈AB〉–〈A〉〈B〉

In[8]:=

variance[s_QuantumState,m_QuantumMeasurementOperator]:=(#〚2〛*(1-#〚2〛))&@Values[m[s]["Probabilities"]]//Ncovariance[s_QuantumState,m1_QuantumMeasurementOperator,m2_QuantumMeasurementOperator]:=(#〚4〛-(#〚3〛+#〚4〛)*(#〚2〛+#〚4〛))&@Values[m2[m1[s]]["Probabilities"]]//N

Implement the first layer circuit and calculate the

(1)

matrix:

In[9]:=

state1=QuantumCircuitOperator"RY"

1,"RY"

2,"RY"

3[];

In[10]:=

g1={{variance[state1,QuantumMeasurementOperator["Z",{1}]],covariance[state1,QuantumMeasurementOperator["Z",{2}],QuantumMeasurementOperator["Z",{1}]]},{covariance[state1,QuantumMeasurementOperator["Z",{1}],QuantumMeasurementOperator["Z",{2}]],variance[state1,QuantumMeasurementOperator["Z",{2}]]}}

Out[10]=

{{0.125,6.93889×

-18

},{6.93889×

-18

,0.1875}}

Second layer circuit diagram:

In[11]:=

QuantumCircuitOperator[{QuantumCircuitOperator[{"RY"[π/4]1,"RY"[π/3]2,"RY"[π/7]3},"Label""\!\(\*SubscriptBox[\(W\), \(0\)]\)"],QuantumCircuitOperator[{"RZ"[0.432],"RZ"[-0.123]2},"Label""\!\(\*SubscriptBox[\(V\), \(1\)]\)"],QuantumCircuitOperator[{"CNOT"{1,2},"CNOT"{2,3}},"Label""\!\(\*SubscriptBox[\(W\), \(1\)]\)"],QuantumMeasurementOperator["PauliY",{1}],QuantumMeasurementOperator["PauliY",{2}]}]["Diagram"]

Out[11]=

Calculate

(2)

using the following formula:

(2)

〈 2 Y 2 〉- 2 〈 Y 2 〉	〈 Y 2 X 3 〉-〈 Y 2 〉〈 X 3 〉
〈 X 3 Y 2 〉-〈 X 3 〉〈 Y 2 〉	 2 X 3 - 2 〈 X 3 〉

Implement the second layer circuit and calculate the

(2)

matrix:

In[12]:=

state2=QuantumCircuitOperator"RY"

1,"RY"

2,"RY"

3,"RZ"[initParameters〚1〛],"RZ"[initParameters〚2〛]2,"CNOT"{1,2},"CNOT"{2,3}[];

In[13]:=

g2={{variance[state2,QuantumMeasurementOperator["Y",{2}]],covariance[state2,QuantumMeasurementOperator["Y",{2}],QuantumMeasurementOperator["X",{3}]]},{covariance[state2,QuantumMeasurementOperator["X",{3}],QuantumMeasurementOperator["Y",{2}]],variance[state2,QuantumMeasurementOperator["X",{3}]]}}

Then the block diagonal matrix is the following:

Alternatively, obtain both parameterized layers automatically:

Calculate the Fubini-Study metric tensor directly:

Define a symbolic and numeric cost function:

Calculate the symbolic cost function gradient:

Compute the quantum natural gradient descent for the symbolic and numeric cost functions using QuantumNaturalGradientDescent:

Next, compute a regular gradient descent for both cost functions using GradientDescent:

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

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

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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 Block Diagonalization

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