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Grover Optimizer#


Grover Adaptive Search (GAS) has been explored as a possible solution for combinatorial optimization problems, alongside variational algorithms such as Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA). The algorithm iteratively applies Grover Search to find the optimum value of an objective function, by using the best-known value from the previous run as a threshold. The adaptive oracle used in GAS recognizes all values above or below the current threshold (for max and min respectively), decreasing the size of the search space every iteration the threshold is updated, until an optimum is found.

In this notebook we will explore each component of the GroverOptimizer, which utilizes the techniques described in GAS, by minimizing a Quadratic Unconstrained Binary Optimization (QUBO) problem, as described in [1]


[1] A. Gilliam, S. Woerner, C. Gonciulea, Grover Adaptive Search for Constrained Polynomial Binary Optimization, arXiv preprint arXiv:1912.04088 (2019).

Find the Minimum of a QUBO Problem using GroverOptimizer#

The following is a toy example of a minimization problem.

\begin{eqnarray} \min_{x \in \{0, 1\}^3} -2x_0x_2 - x_1x_2 - 1x_0 + 2x_1 - 3x_2. \end{eqnarray}

For our initial steps, we create a docplex model that defines the problem above, and then use the from_docplex_mp() function to convert the model to a QuadraticProgram, which can be used to represent a QUBO in Qiskit Optimization.

from qiskit_algorithms import NumPyMinimumEigensolver
from qiskit.primitives import Sampler
from qiskit_optimization.algorithms import GroverOptimizer, MinimumEigenOptimizer
from qiskit_optimization.translators import from_docplex_mp
from import Model
model = Model()
x0 = model.binary_var(name="x0")
x1 = model.binary_var(name="x1")
x2 = model.binary_var(name="x2")
model.minimize(-x0 + 2 * x1 - 3 * x2 - 2 * x0 * x2 - 1 * x1 * x2)
qp = from_docplex_mp(model)
Problem name: docplex_model1

  -2*x0*x2 - x1*x2 - x0 + 2*x1 - 3*x2

Subject to
  No constraints

  Binary variables (3)
    x0 x1 x2

Next, we create a GroverOptimizer that uses 6 qubits to encode the value, and will terminate after there have been 10 iterations of GAS without progress (i.e. the value of \(y\) does not change). The solve() function takes the QuadraticProgram we created earlier, and returns a results object that contains information about the run.

grover_optimizer = GroverOptimizer(6, num_iterations=10, sampler=Sampler())
results = grover_optimizer.solve(qp)
objective function value: -6.0
variable values: x0=1.0, x1=0.0, x2=1.0
status: SUCCESS

This results in the optimal solution \(x_0=1\), \(x_1=0\), \(x_2=1\) and the optimal objective value of \(-6\) (most of the time, since it is a randomized algorithm). In the following, a custom visualization of the quantum state shows a possible run of GroverOptimizer applied to this QUBO.


Each graph shows a single iteration of GAS, with the current values of \(r\) (= iteration counter) and \(y\) (= threshold/offset) shown in the title. The X-axis displays the integer equivalent of the input (e.g. ‘101’ \(\rightarrow\) 5), and the Y-axis shows the possible function values. As there are 3 binary variables, there are \(2^3=8\) possible solutions, which are shown in each graph. The color intensity indicates the probability of measuring a certain result (with bright intensity being the highest), while the actual color indicates the corresponding phase (see phase color-wheel below). Note that as \(y\) decreases, we shift all of the values up by that amount, meaning there are fewer and fewer negative values in the distribution, until only one remains (the minimum).


Check that GroverOptimizer finds the correct value#

We can verify that the algorithm is working correctly using the MinimumEigenOptimizer in Qiskit optimization.

exact_solver = MinimumEigenOptimizer(NumPyMinimumEigensolver())
exact_result = exact_solver.solve(qp)
objective function value: -6.0
variable values: x0=1.0, x1=0.0, x2=1.0
status: SUCCESS
import tutorial_magics


Version Information

System information
Python version3.8.18
Wed Feb 28 02:58:05 2024 UTC

This code is a part of a Qiskit project

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