Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Quantum Optimization
In this Tech Note, we document the implementation and utilization of essential functions used in the for Quantum Optimization algorithms.
By providing a comprehensive overview and usage guidelines for these functions, we aim to introduce new and experienced users into quantum optimization techniques 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.
Details and Options
Details and Options
QuantumLinearSolve works on numerical matrices. The argument b must be a vector, while the argument m must be a square matrix. QuantumLinearSolve is able to compute solutions for –dimension problems.
n
2
QuantumLinearSolve utilizes a multiplexer-based variational quantum linear solver algorithm. The primary distinctions from a conventional variational quantum linear solver algorithm are as follows:
◼
The multiplication of the matrix is implemented by a multiplexer, enabling the accurate representation of the operation within the quantum circuit.
m
A.x
◼
The solution to the linear system is encoded directly in the amplitudes of the resultant quantum state.
This approach simplifies the standard variational quantum linear solver by reducing the need for multiple circuits used in the real-imaginary decomposition of the solution and the term-by-term computation within quantum circuits. A detailed step-by-step implementation is outlined in this documentation.
The basic steps to implement the algorithm are the following:
◼
Start with a variational circuit ansatz or directly an state ansatz
V()
ω
i
x()=V()|0〉
ω
i
ω
i
◼
Apply the multiplexer to perform and obtain resultant state
m.x()
ω
i
ψ()
ω
i
◼
Use state as the input for the optimizer with the cost function to obtain new parameters
ψ()
ω
i
f(ω)=1-
b
ψ(ω)
2
ω
j
◼
Use new parameters with the variational circuit or state ansatz
ω
j
V()
ω
i
x()
ω
j
◼
Repeat the process until you obtain
f()=1-=0
ω
opt
b
ψ()
ω
opt
2
Algorithm Implementation
Algorithm Implementation
Ansatz
Multiplexer
Cost Function
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
GlobalPhaseAccuracy
AccuracyGoal & PrecisionGoal
Method
Heuristic methods include:
"NelderMead" | use only convex methods |
"DifferentialEvolution" | use differential evolution |
"SimulatedAnnealing" | use simulated annealing |
"RandomSearch" | use the best local minimum found from multiple random starting points |
"Couenne" | use the Couenne library for non-convex mixed-integer nonlinear problems |
In[10]:=
m=RandomReal[{0,1},{4,4}]
Out[10]=
{{0.00393484,0.726178,0.647708,0.486648},{0.425836,0.192925,0.135447,0.156309},{0.0473378,0.134998,0.214507,0.380623},{0.916129,0.637982,0.761709,0.375096}}
In[11]:=
b=RandomReal[{0,1},4]
Out[11]=
{0.383922,0.335146,0.175575,0.653389}
In[12]:=
QuantumLinearSolve[m,b,Method"DifferentialEvolution"]
Out[12]=
{0.477453+0.,0.67417+0.,-0.500057+0.,0.444606+0.}
Some methods may give suboptimal results for certain problems:
In[13]:=
solutions=(#QuantumLinearSolve[m,b,Method#])&/@{Automatic,"NelderMead","DifferentialEvolution","SimulatedAnnealing"};
::error
In[14]:=
solutions//TableForm
Out[14]//TableForm=
Automatic{0.477453+0.,0.67417+0.,-0.500057+0.,0.444606+0.} |
NelderMead{0.477453+0.,0.674175+0.,-0.500064+0.,0.444607+0.} |
DifferentialEvolution{0.477453+0.,0.67417+0.,-0.500057+0.,0.444606+0.} |
SimulatedAnnealing{0.477453+0.,0.674171+0.,-0.500058+0.,0.444607+0.} |
WorkingPrecision
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.
Compare the results:
Properties
You can request the components used during the calculation using a third property argument:
Show the ansatz quantum state used for parameter optimization"
Show the quantum circuit used for parameter optimization:
Request more than one property:
Examples Custom Functions