I implemented the combination optimization problem in blueqat before (see [here] for the article on blueqat [https://qiita.com/Tusnori/items/71ea8d01fb4892e0eae9)), but this time I would like to implement it in Qiskit. Since this article is intended to confirm the implementation method, I would like to deal with the simple combination optimization problem itself.

Problem to solve

The problem to solve is the same as before. Find the combination of $ q (0) $ and $ q (1) $ that minimizes the following Hamiltonian $ H . $ H = 1 - q(0) - q(1) $$ Note that $ q (0) $ and $ q (1) $ are either 0 or 1. In this problem setting, Hamiltonian $ H = -1 $ is the minimum value when $ q (0) = q (1) = 1 $.

Supplementary information

By the way, Hamiltonian has the following values for all combinations of $ q (0) and q (1) $.

q(0)=0,q(1)=0 \rightarrow H = 1
q(0)=0,q(1)=1 \rightarrow H = 0
q(0)=1,q(1)=0 \rightarrow H = 0
q(0)=1,q(1)=1 \rightarrow H = -1

About the execution environment

Run with Google Colabratory. The version of Qiskit is "0.14.1".

Source code

Install Qiskit

pip install qiskit

Import required packages for Qiskit

from qiskit import BasicAer
from qiskit.aqua.algorithms import QAOA
from qiskit.optimization.algorithms import MinimumEigenOptimizer
from qiskit.optimization import QuadraticProgram

Definition of variable, Hamiltonian

qubo = QuadraticProgram()
#Binary variable definition
qubo.binary_var('q0')
qubo.binary_var('q1')
#Hamiltonian definition
qubo.minimize(linear=[-1,-1],constant=1.0)
print(qubo.export_as_lp_string())

The definition result of the variable Hamiltonian is as follows.

\ This file has been generated by DOcplex
\ ENCODING=ISO-8859-1
\Problem name: CPLEX

Minimize
 obj: - q0 - q1 + 1
Subject To

Bounds
 0 <= q0 <= 1
 0 <= q1 <= 1

Binaries
 q0 q1
End

Solve the Hamiltonian defined in QAOA.

#Initialization of MinimumEigensolver
qaoa_mes = QAOA(quantum_instance=BasicAer.get_backend('statevector_simulator'))

#Creating an Optimizer
qaoa = MinimumEigenOptimizer(qaoa_mes) 

#QAOA calculation/Result output
qaoa_result = qaoa.solve(qubo)
print("QAOA result:")
print(qaoa_result)

The execution result is as follows.

QAOA result:
x=[1.0,1.0], fval=-1.0

I was able to find a solution successfully!

Impressions of implementing

The impression I implemented is "difficult to write ...". What is particularly strange is the Hamiltonian definition. I thought I could handle the binary variables q0 and q1 like Sympy's Symbol, but that doesn't seem to be the case.

Finally

Please let me know if there is a better implementation in Qiskit. I personally think it's not very smart.