Introduction to QUBO

Behzad Mansouri

This article consists of a short introduction to the quantum approximation optimisation algorithm (QAOA). The mathematical structure of the QAOA, as well as its basic properties, are described. The implementation of the QAOA on MaxCut problems, quadratic unconstrained binary optimisation problems (QUBOs), and Ising-type Hamiltonians is considered in detail.

Thorsten Koch, Daniel Rehfeldt, Yuji Shinano

Karthik Natarajan

Michał Jaroszczuk

Vehicle Routing Problem is a well-known optimization issue, which has many extensions resulting from practical appliances. The problem is NP-hard, therefore it may be impossible to find an optimal solution for big instances, with traditional computation methods. Recent development of quantum technology, may be a response for this challenge, which however requires a different formulation of the model. The problem is then modeled using Quadratic Unconstrained Binary Optimization (QUBO) equations. Such model can be run using quantum annealers (QA), designed specifically for optimization purposes. This paper is a systematic survey of Vehicle Routing Problem models that are formulated using QUBO, to be solved on quantum computers. The main goal of the paper is to check research gaps in this subject, compare and explain different approaches of modeling various problem extensions with QUBO. The results obtained in reviewed papers will be presented and analyzed. Moreover, the theoretical basis for quantum computation will be presented, with focus on D-Wave company solutions.

A.M. Semenov, S.R. Usmanov, Aleksey K. Fedorov

Practical discrete optimization problems often contain multidimensional arrays of variables with linear constraints, which complicates their conversion to QUBO (quadratic unconstrained binary optimization) form. The article proposes a systematic approach to transforming such problems, which includes three key steps: the transition from a multidimensional representation of variables to a one-dimensional one using the Kronecker product of matrices, the reduction of mixed variables to binary ones, and the introduction of linear constraints into the objective function through quadratic penalties. Explicit computational formulas are obtained for each stage, simplifying their software implementation. The developed method is illustrated with examples from graph theory and combinatorial optimization, including classical formulations, confirming its universality. The results of the article allow standardizing the process of adapting problems for solving on quantum annealing algorithms (e.g., D-Wave) and classical QUBO solvers.

Abraham P. Punnen