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In this Tech Note, we document the implementation and utilization of essential functions used in the for Quantum Computing algorithms. The examples include the Quantum Natural Gradient Descent, Stochastic Parameter Shift Rule and more.
By providing a comprehensive overview and usage guidelines for these functions, we aim to introduce new and experienced users into quantum optimization techniques, quantum machine learning and quantum computing research.
In[92]:=
<<Wolfram`QuantumFramework`ExampleRepository`
Gradient-Based Optimization Methods
Quantum Linear Solver
The Variational Quantum Linear Solver (VQLS) is a hybrid quantum-classical technique designed to address Quantum Linear Systems Problems (QLSP). The main goal is to find linear solutions for systems as .
m.x=b
In this Tech Note we will explain the applications and options of function implemented in the Wolfram Quantum Computation Framework to solve QLSP problems.
QuantumLinearSolve
QuantumLinearSolve | uses a hybrid optimization algorithm to find a quantum state described by the state vector x m.x=b |
QuantumLinearSolve | solves m.x=b prop |
We will briefly demonstrate how to apply all these functions in Wolfram quantum framework.
Example
Example
Simple example
Generate a random 4×4 real matrix:
In[93]:=
m=RandomReal[{0,1},{4,4}];m//MatrixForm
Out[94]//MatrixForm=
0.596166 | 0.62566 | 0.816575 | 0.857365 |
0.287865 | 0.236078 | 0.491353 | 0.593494 |
0.786389 | 0.0297055 | 0.638947 | 0.00472644 |
0.736549 | 0.108758 | 0.314166 | 0.0812839 |
Generate a random complex vector of the length 4:
In[95]:=
b=RandomComplex[{-0.1-0.1I,0.1+0.1I},4]
Out[95]=
{0.0170825+0.049994,0.027093+0.091541,0.0806183-0.0000258573,-0.0829871-0.0489408}
Use them as input in the :
QuantumLinearSolve
In[99]:=
QuantumLinearSolve[m,b]
Out[99]=
{-0.287755-0.110093,-0.0772302-0.274276,0.485296+0.146765,-0.185835+0.195234}
When running above code, you may see a progress box describing steps and estimated time.
In[100]:=
LinearSolve[m,b]
Out[100]=
{-0.287755-0.110093,-0.0772302-0.274276,0.485296+0.146764,-0.185835+0.195234}
Compare the results:
In[98]:=
%%-%//Abs
Out[98]=
{4.17675×,2.38047×,5.30811×,1.74998×}
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10
-7
10
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10
-7
10
Properties
You can request the components used during the calculation using a third property argument:
"Ansatz" | Returns only the QuantumState ansatz used to solve the QLSP. |
"CircuitOperator" | Returns only the QuantumCircuitOperator variational circuit ψ used to solve the QLSP. |
"GlobalPhase" | Includes in solution the calculated global phase 〈 ϕ |
"OptimizedParameters" | Return optimized values of defined parameters. |
"Parameters" | Return systems defined parameters. |
Show the ansatz quantum state used for parameter optimization"
In[101]:=
QuantumLinearSolve[m,b,"Ansatz"]
Out[101]=
QuantumState
 |  |
|
Show the quantum circuit used for parameter optimization:
In[102]:=
QuantumLinearSolve[m,b,"CircuitOperator"]
Out[102]=
Request more than one property:
In[103]:=
QuantumLinearSolve[m,b,{"Result","Ansatz","GlobalPhase"}]
Out[103]=
Result{-0.287755-0.110093,-0.0772302-0.274275,0.485296+0.146765,-0.185835+0.195234},AnsatzQuantumState
,GlobalPhase
|  |
|
-0.23026347
-0.46098513
±
0.00000014
In[6]:=
QuantumLinearSolve[m,b,All]
Out[6]=
Result{0.0594845+0.0894924,-0.420933-0.290229,-0.289422-0.169629,0.544499+0.292522},AnsatzQuantumState
,CircuitOperatorQuantumCircuitOperator
,GlobalPhase
|  |
|
|  |
|
-0.20612061
+0.43273420
±
0.00000005
,OptimizedParameters{ω.$1-0.694518,ω.$22.89257,ω.$31.81246,ω.$4-3.25301,ω.$50.0993645,ω.$6-1.6663,ω.$7-1.22972,ω.$82.38821},Parameters{ω.$1,ω.$2,ω.$3,ω.$4,ω.$5,ω.$6,ω.$7,ω.$8}Options
Options
"Ansatz" | Automatic | use specified ansatz instead of the one generated by the function |
"GlobalPhaseAccuracy" | -5 10 | expected accuracy for global phase estimation |
Automatic | number of digits of final accuracy sought | |
Automatic | maximum number of iterations to use | |
Automatic | method to use | |
Automatic | number of digits of final precision sought | |
the precision used in internal computations |
Ansatz
In[15]:=
ansatz=
[{α1,α2,α3,α4},{1,2},"Parameters"{α1,α2,α3,α4}];ansatz["Formula"]
 | QuantumState |
Out[16]=
α1|00〉+α2|01〉+α3|10〉+α4|11〉
In[17]:=
m=RandomReal[{0,1},{4,4}]
Out[17]=
{{0.251149,0.632406,0.00760061,0.451067},{0.768939,0.771227,0.223159,0.103032},{0.553352,0.929828,0.0607338,0.399935},{0.778583,0.8705,0.598142,0.676169}}
In[18]:=
b=RandomReal[{0,1},4]
Out[18]=
{0.732115,0.567505,0.576803,0.848305}
In[19]:=
QuantumLinearSolve[m,b,"Ansatz"ansatz]
Out[19]=
{12.147+0.,-9.61246+0.,-10.0188+0.,8.50546+0.}
Compare the result with the classical method:
In[20]:=
LinearSolve[m,b]
Out[20]=
{12.147,-9.61247,-10.0188,8.50547}
In[21]:=
%-%%//Abs
Out[21]=
{2.77119×,2.35603×,2.31051×,1.90121×}
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10
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10
GlobalPhaseAccuracy
The function does an approximation to recover the result once the variational circuit ψ(ω) is close enough to the vector problem such that . "GlobalPhaseAccuracy" option sets the accuracy of the global phase calculated.
QuantumLinearSolve
ψ≃
iϕ
In[22]:=
m=RandomReal[{0,1},{4,4}]
Let's indicate a high accuracy:
AccuracyGoal & PrecisionGoal
We can change them in order to get less precision but faster timing:
Compare both approximate results with the original result:
Method
Methods that are guaranteed to give a global minimum when they converge to a solution include:
Heuristic methods include:
Some methods may give suboptimal results for certain problems:
WorkingPrecision
Examples Custom Functions