The Stochastic Parameter–Shift Rule

Differentiate qubit gates using the exact and approximate stochastic parameter–shift rule

The expected value of a circuit in relation to a measurement operator

varies smoothly with the circuit's gate parameters θ. This implies that the derivatives

∇

〈A〉

are well–defined, allowing for the application of gradient descent in any unitary gate of the form

iθG

We first express

G=H+θV

, where

is a tensor product of Pauli operators and H represents any linear combination of tensor products involving Pauli operators, chosen arbitrarily.

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`"]

The stochastic parametric–shift rule (SPSR) uses the following 3 gates:

In[4]:=

QuantumCircuitOperator"0",QuantumCircuitOperatorQuantumOperator"RandomUnitary","Label""

θ(1-s)G



",QuantumOperator"RandomUnitary","Label""

±ϕV



",QuantumOperator"RandomUnitary","Label""

θsG



","Label""SPSR Gates",QuantumMeasurementOperator["Computational","Label""A"]["Diagram","ShowMeasurementWire"False,ImageSizeLarge,"WireLabels"{Placed[Style["〈

〉",FontSize20],Right]}]

Out[4]=

where s is a uniform random variable in the interval

[0,1]

, which we will be sampling multiple times for each θ. The expected value of this circuit will be called

〈

〉

〈

〉

depending on the ϕ shift sign. The gradient can be obtained from the average:

∇

〈A〉=

sϵ[0,1]

[〈

〉-〈

〉]

Test the SPSR algorithm using a Cross–Resonance gate:

(

)=exp[(

X-

ZX+

X)]

Implement the generator and the gate:

In[5]:=

CrossResonanceGenerator=QuantumOperator[(θ1*QuantumOperator[{"X"{1},"I"{2}}]-θ2*QuantumOperator[{"Z"{1},"X"{2}}]+θ3*QuantumOperator[{"I"{1},"X"{2}}]),"Parameters"{θ1,θ2,θ3}];

In[6]:=

CrossResonanceGate=QuantumCircuitOperator[{Exp[I*CrossResonanceGenerator],QuantumMeasurementOperator["PauliZ"{1,-1}]},"Parameters"{θ1,θ2,θ3}];

Fix

and

and differentiate with respect to

=θ

(

V=X

). The correspondent stochastic parameter–shift rule circuit is given by:

In[7]:=

SPSRCircuit=QuantumCircuitOperator[{QuantumOperator[Exp[(1-s)*I*CrossResonanceGenerator[θ20.15,θ31.6]]],QuantumOperator[Exp[I*ϕ*"PauliX"]],QuantumOperator[Exp[s*I*CrossResonanceGenerator[θ20.15,θ31.6]]],QuantumMeasurementOperator["Z"{1,-1}]},"Parameters"{ϕ,θ1,s}];

In[8]:=

SPSRCircuit[ϕ±ϕ]["Diagram",ImageSizeLarge]

Out[8]=

As an example, calculate the gradient for

In[9]:=

MeanTableSPSRCircuitθ1π4.,srandomValue,ϕ

[]["Mean"]-SPSRCircuitθ1π4.,srandomValue,ϕ-

[]["Mean"],{randomValue,RandomReal[1,5]}

Out[9]=

-1.98429

Use the function SPSRGradientValues to compute

∇

〈A〉

. It receives as input the matrix generator as a function of one variable (i.e.

) and the Pauli operator associated with that variable:

In[10]:=

CrossResonanceMatrix[θ1_]=CrossResonanceGenerator[θ20.15,θ31.6]["Matrix"]//Normal

Out[10]=

{{0,1.45,θ1,0},{1.45,0,0,θ1},{θ1,0,0,1.75},{0,θ1,1.75,0}}

In[11]:=

CrossResonanceGradient=SPSRGradientValues[CrossResonanceMatrix,"PauliX"];//AbsoluteTiming

Out[11]=

{47.9826,Null}

Compare the cross resonance gradient with the correspondent mean value:

In[12]:=

CrossResonanceMean=Table[{θ,#/.θ1θ},{θ,Subdivide[0.0001,2.π,50]}]&@CrossResonanceGate[θ20.15,θ31.6][]["Mean"];//AbsoluteTiming

Out[12]=

{1.22135,Null}

In[13]:=

ListLinePlot{CrossResonanceGradient,CrossResonanceMean},



Out[13]=

	Stochastic parameter-shift rule gradient
	Mean value

Find analytical solutions using Fourier interpolation:

In[14]:=

Fit[CrossResonanceMean,Table[Exp[nθ],{n,-5,5}],θ]//Re//ComplexExpand//Chop[#,

-2

Out[14]=

0.995065Cos[2θ]

In[15]:=

Fit[CrossResonanceGradient,Table[Exp[nθ],{n,-5,5}],θ]//Re//ComplexExpand//Chop[#,

-2

Out[15]=

-1.99022Sin[2θ]

The implementation of gates with form

±



for the SPSR algorithm could pose a challenge for some hardware. In order to address that problem we can implement the Approximate Stochastic Parameter-shift rule using the following 3 gates:

In[16]:=

QuantumCircuitOperator"0",QuantumCircuitOperatorQuantumOperator"RandomUnitary","Label""

θ(1-s)G



",QuantumOperator"RandomUnitary","Label""

ϵH±

4ϵ

ϕV



",QuantumOperator"RandomUnitary","Label""

θsG



","Label""ASPSR Gates",QuantumMeasurementOperator["Computational","Label""A"]["Diagram","ShowMeasurementWire"False,ImageSizeLarge,"WireLabels"{Placed[Style["〈

〉",FontSize20],Right]}]

Out[16]=

Implement the

operator from our generator:

In[17]:=

H=-0.15*QuantumOperator[{"Z"{1},"X"{2}}]+1.6*QuantumOperator[{"I"{1},"X"{2}}];

Use the function ASPSRGradientValues to compute

∇

〈A〉

. It receives as input the matrix generator as a function of one variable (i.e.

) and the Pauli operator associated with that variable and the H operator:

In[18]:=

CrossResonanceMatrix[θ1_]=CrossResonanceGenerator[θ20.15,θ31.6]["Matrix"]//Normal

Out[18]=

{{0,1.45,θ1,0},{1.45,0,0,θ1},{θ1,0,0,1.75},{0,θ1,1.75,0}}

Compare the results from both the exact and approximate SPSR algorithm:

Find analytical solutions using Fourier interpolation and compare it with the exact result:

External Links

Source Metadata

Citation:
- Parameter Shift-Rule to Differentiate Qubit Gates: Stochastic and Approximate Stochastic Algorithms
  by Sebastian Rodriguez
  Wolfram Community, STAFF PICKS, February 28, 2024
  https://community.wolfram.com/groups/-/m/t/3132207
- Killoran, N. (2020, May 24). The stochastic parameter-shift rule. PennyLane Demos. https://pennylane.ai/qml/demos/tutorial_stochastic_parameter _shift/
- Banchi, L., & Crooks, G. E. (2021). Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule. In Quantum (Vol. 5, p. 386). Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2021-01-25-386

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Contributed by: Wolfram Research, Quantum Computation Framework Team

Wolfram Language Example Repository

The Stochastic Parameter–Shift Rule

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