Quantum annealing computers are designed to produce high-quality solutions to optimization problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. While most of the well known NP-hard problems can easily be represented as quadratic binary problems, such formulations usually contain constraints that have to be added as penalties to the objective function in order to obtain QUBOs. In this paper, we propose a method based on finite automaton representation of the constraints for generating penalty implementations for them, which uses fewer qubits than the alternatives and is general enough to be applied to a whole class of constraints.
Quantum annealing computers are designed to produce high-quality solutions to optimization problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. While most of the well known NP-hard problems can easily be represented as quadratic binary problems, such formulations usually contain constraints that have to be added as penalties to the objective function in order to obtain QUBOs. In this paper, we propose a method based on finite automaton representation of the constraints for generating penalty implementations for them, which uses fewer qubits than the alternatives and is general enough to be applied to a whole class of constraints.
Quantum annealing is a model for quantum computing that is aimed at solving hard optimization problems by representing them as quadratic unconstrained binary optimization (QUBO) problems. Although many NP-hard problems can easily be formulated as binary-variable problems with a quadratic objective function, such formulations typically also include constraints, which are not allowed in a QUBO. Hence, such constraints are usually incorporated in the objective function as additive penalty terms. While there is substantial previous work on implementing linear equality constraints, the case of inequality constraints has not much been studied. In this paper, we propose a new approach for formulating and embedding inequality constraints as penalties and describe early implementation results.
Quantum annealers such as the D-Wave 2X computer are designed to natively solve Quadratic Unconstrained Binary Optimization (QUBO) problems to optimality or near optimality. Most NP-hard problems, which are hard for classical computers, can be naturally described as quadratic binary problems that contain a quadratic binary objective function and one or more constraints. Since a QUBO cannot have constraints, each such constraint has to be added to the objective function as a penalty, in order to solve on D-Wave. For a minimization problem, for instance, such penalty can be a quadratic term that gets a value zero, if the constraint is satisfied, and a large value, if it is not. In many cases, however, the penalty can significantly increase the number of quadratic terms in the resulting QUBO and make it too large to embed into the D-Wave hardware. In this paper, we develop an alternative method for formulating and embedding constraints of the type Σi=1sxi=1, which is much more scalable than the existing ones, and analyze the properties of the resulting embeddings.
Quantum annealers such as the commercially available D-Wave machines are designed to natively solve quadratic unconstrained binary optimization (QUBO) problems. While most of the well-known NP-hard optimization problems can easily be formulated as quadratic binary problems, such formulations also contain constraints, which commonly are added to the objective function in the form of penalties to obtain a QUBO version. However, the standard method for defining such penalties leads to QUBOs that are dense and therefore take too much of the resources of the quantum annealer. In this paper, we describe an alternative approach to the constraint embedding problem that uses mixed-integer linear programming (MILP) and is scalable to problems of arbitrary number of variables.
This paper proposes a novel framework for automatically setting penalty weights in the Quadratic Unconstrained Binary Optimization (QUBO) to handle problems with multiple constraints. Quantum computing and quantum-inspired methods have gained prominence, with QUBO models widely used for their unified representation of optimization problems. However, QUBO performance is highly sensitive to penalty weight settings, especially in multiple constraints scenarios, where setting different penalty weights to different constraints may further affect performance. Existing methods for setting common penalty weights for all constraints in QUBO can be generally categorized into manual methods based on expert experience, exact methods based on the objective function values, and sequential methods based on searching within an upper bound. This study introduces a novel framework for multiple constraints optimization that utilizes exact and sequential binary search methods to determine common penalty weight while integrating constraint sensitivity analysis to adjust each penalty weight dynamically. This approach enhances the efficiency and quality of QUBO solutions. Experimental results on the Travelling Salesman Problem show that the proposed framework achieves superior solution quality and feasibility on both the D-Wave quantum annealer and classical simulated annealing solvers, outperforming traditional methods and demonstrating strong adaptability.
Quantum annealers offer an efficient way to compute high quality solutions of NP-hard problems when expressed in a QUBO (quadratic unconstrained binary optimization) or an Ising form. This is done by mapping a problem onto the physical qubits and couplers of the quantum chip, from which a solution is read after a process called quantum annealing. However, this process is subject to multiple sources of biases, including poor calibration, leakage between adjacent qubits, control biases, etc., which might negatively influence the quality of the annealing results. In this work, we aim at mitigating the effect of such biases for solving constrained optimization problems, by offering a two-step method, and apply it to Graph Partitioning. In the first step, we measure and reduce any biases that result from implementing the constraints of the problem. In the second, we add the objective function to the resulting bias-corrected implementation of the constraints, and send the problem to the quantum annealer. We apply this concept to Graph Partitioning, an important NP-hard problem, which asks to find a partition of the vertices of a graph that is balanced (the constraint) and minimizes the cut size (the objective). We first quantify the bias of the implementation of the constraint on the quantum annealer, that is, we require, in an unbiased implementation, that any two vertices have the same likelihood of being assigned to the same or to different parts of the partition. We then propose an iterative method to correct any such biases. We demonstrate that, after adding the objective, solving the resulting bias-corrected Ising problem on the quantum annealer results in a higher solution accuracy.
Quantum annealing is an optimization technique that uses quantum fluctuation effects to search for solutions and is being applied as a metaheuristic method. Quantum annealing solves a problem expressed as quadratic unconstrained binary optimization (QUBO). Many practical problems have complex constraints, and formulating them can lead to third-order or higher expressions. In such cases, higher-order expressions have been converted to QUBO by order reduction. Various order reduction methods have been used, but they have issues, such as the need to add more auxiliary variables and difficulty in obtaining optimal solutions. This paper discusses a method for formulating problems with complex constraints using inequalities. In this approach, each constraint is defined as a linear function and added together to form an inequality. The inequality is then transformed to QUBO by introducing a slack variable. The application to the quantum minimum fill-in algorithm, a fill-in reduction ordering algorithm for sparse matrices, and the maximum satisfiability problem are also presented. The results of solving these problems with a quantum-inspired annealing machine show that higher-quality solutions are obtained with fewer variables than when using higher-order expressions with order reduction.