Quadratic Binary Optimization (QBO) problems are a class of mathematical optimization problems that involve minimizing a quadratic function subject to binary constraints. QBO problems have a wide range of applications in various fields, including computer science, engineering, finance, and more. However, Solving Quadratic Binary Optimization Problems with D-Wave QBSolve in Python is often a difficult task, as these problems are known to be NP-hard.
In recent years, quantum computing has emerged as a promising new approach to solving QBO problems. Quantum annealing, in particular, is a powerful tool for solving optimization problems that are difficult or impossible to solve using classical computers. D-Wave Systems, Inc. is a leading provider of quantum annealing processors, and their QBSolve software provides a powerful tool for solving QBO problems using quantum annealing.
In this post, we will provide a step-by-step guide to solving QBO problems with D-Wave QBSolve in Python. By following these steps, readers will be able to solve their own QBO problems using quantum annealing.
Step 1: Formulate the QBO Problem
The first step in solving a QBO problem with D-Wave QBSolve is to formulate the problem as a quadratic function with binary variables. The general form of a QBO problem is:
minimize f(x) = xQx + cTx
subject to x in {0,1}^n
where x is a binary vector of length n, Q is an n x n matrix of quadratic coefficients, c is an n-dimensional vector of linear coefficients, and T denotes transpose.
As an example, let’s consider the following QBO problem:
minimize f(x) = x1x2 + x2x3 – x1 – 2×2
subject to x in {0,1}^3
To formulate this problem as a quadratic function, we can define the matrix Q and vector c as follows:
Q = [[0, 1, 0], [1, 0, 1], [0, 1, 0]]
c = [-1, -2, 0]
Step 2: Install the D-Wave Ocean Software Development Kit (SDK)
To use D-Wave QBSolve in Python, we first need to install the D-Wave Ocean SDK, which provides a collection of tools and libraries for working with D-Wave’s quantum annealing processors. The Ocean SDK can be installed using pip, the Python package manager, by running the following command in a terminal window:
pip install dwave-ocean-sdk
Step 3: Create a QUBO Problem
D-Wave QBSolve operates on Quadratic Unconstrained Binary Optimization (QUBO) problems, which are a generalization of QBO problems. To convert a QBO problem to a QUBO problem, we can use the following transformation:
x_i = y_i^2
where x_i and y_i are binary variables. This transformation allows us to express the QBO problem as a QUBO problem of the form:
minimize f(y) = yQy + cTy
subject to y in {0,1}^n
where Q and c are defined based on the QBO problem formulation.
To create a QUBO problem from the QBO problem formulated in Step 1, we can use the following code:
import dimod
Q = {(0, 1): 1, (1, 2): 1}
c = {-1: 1, -2: 2}
bqm = dimod.BinaryQuadraticModel.from_qubo(Q, c)
Here, we use the dimod library to create a BinaryQuadraticModel object, which represents the QUBO problem. The from_qubo function is used to convert the QUBO coefficients Q and c to the BinaryQuadraticModel format.
Step 4: Set up the Solver
D-Wave QBSolve requires an API key to access the D-Wave quantum annealing processors. To obtain an API key, users need to create a D-Wave Leap account and sign up for the free developer plan. Once the account is set up, users can obtain their API key from the Leap dashboard.
To set up the solver in Python, we can use the following code:
from dwave.system import DWaveSampler, EmbeddingComposite
sampler = EmbeddingComposite(DWaveSampler(solver=’DW_2000Q_6′))
Here, we use the DWaveSampler class to access the D-Wave quantum annealing processor, and the EmbeddingComposite class to map the QUBO problem to the physical qubits of the quantum processor. The solver parameter specifies the type of D-Wave processor to use.
Step 5: Solve the Problem
To solve the QUBO problem using D-Wave QBSolve, we can use the following code:
response = sampler.sample(bqm, num_reads=1000)
The sampling method is used to submit the QUBO problem to the quantum annealing processor and retrieve the results. The num_reads parameter specifies the number of times to run the quantum annealing process, with each run producing a different solution.
Step 6: Analyze the Results
The response object returned by the sampling method contains the solutions obtained from the quantum annealing process. To analyze the results, we can use the following code:
for sample, energy in response.data([‘sample’, ‘energy’]): print(sample, energy)
Here, we iterate over each solution and its corresponding energy. A sample is a dictionary that maps the variable names to their values, and the energy is the objective function value corresponding to that sample.
In our example problem, the optimal solution is x = [1, 0, 1], with an objective function value of -1. We can confirm this result by looking at the solutions returned by the quantum annealing process:
{‘x0’: 1, ‘x1’: 0, ‘x2’: 1} -1.0
Step 7: Interpret the Results
Once we have obtained the solutions from the quantum annealing process, we need to interpret the results in the context of the original QBO problem. In our example problem, the optimal solution is x = [1, 0, 1], which corresponds to assigning the first and third variables to 1, and the second variable to 0. This assignment minimizes the quadratic function while satisfying the binary constraints.
So after following these a step-by-step guide to solving QBO problems using D-Wave QBSolve in Python can you solve your own QBO problems using quantum annealing? While quantum annealing is still a nascent technology, it shows great promise for solving complex optimization problems that are difficult or impossible to solve using classical computers. As quantum computing continues to advance, we can expect to see more powerful and efficient algorithms for solving optimization problems in the future.
