Quadratic Unconstrained Binary Optimization Problem Research Articles

Capacitated Clustering via Majorization-Minimization and Collaborative Neurodynamic Optimization.

This paper addresses capacitated clustering based on majorization-minimization and collaborative neurodynamic optimization (CNO). Capacitated clustering is formulated as a combinatorial optimization problem. Its objective function consists of fractional terms with intra-cluster similarities in their numerators and cluster cardinalities in their denominators as normalized cluster compactness measures. To obviate the difficulty in optimizing the objective function with factional terms, the combinatorial optimization problem is reformulated as an iteratively reweighted quadratic unconstrained binary optimization problem with a surrogate function and a penalty function in a majorization-minimization framework. A clustering algorithm is developed based on CNO for solving the reformulated problem. It employs multiple Boltzmann machines operating concurrently for local searches and a particle swarm optimization rule for repositioning neuronal states upon their local convergence. Experimental results on ten benchmark datasets are elaborated to demonstrate the superior clustering performance of the proposed approaches against seven baseline algorithms in terms of 21 internal cluster validity criteria.

QUBO Problem Formulation of Fragment-Based Protein-Ligand Flexible Docking.

Protein-ligand docking plays a significant role in structure-based drug discovery. This methodology aims to estimate the binding mode and binding free energy between the drug-targeted protein and candidate chemical compounds, utilizing protein tertiary structure information. Reformulation of this docking as a quadratic unconstrained binary optimization (QUBO) problem to obtain solutions via quantum annealing has been attempted. However, previous studies did not consider the internal degrees of freedom of the compound that is mandatory and essential. In this study, we formulated fragment-based protein-ligand flexible docking, considering the internal degrees of freedom of the compound by focusing on fragments (rigid chemical substructures of compounds) as a QUBO problem. We introduced four factors essential for fragment-based docking in the Hamiltonian: (1) interaction energy between the target protein and each fragment, (2) clashes between fragments, (3) covalent bonds between fragments, and (4) the constraint that each fragment of the compound is selected for a single placement. We also implemented a proof-of-concept system and conducted redocking for the protein-compound complex structure of Aldose reductase (a drug target protein) using SQBM+, which is a simulated quantum annealer. The predicted binding pose reconstructed from the best solution was near-native (RMSD = 1.26 Å), which can be further improved (RMSD = 0.27 Å) using conventional energy minimization. The results indicate the validity of our QUBO problem formulation.

Binary matrix factorization via collaborative neurodynamic optimization

Binary matrix factorization is an important tool for dimension reduction for high-dimensional datasets with binary attributes and has been successfully applied in numerous areas. This paper presents a collaborative neurodynamic optimization approach to binary matrix factorization based on the original combinatorial optimization problem formulation and quadratic unconstrained binary optimization problem reformulations. The proposed approach employs multiple discrete Hopfield networks operating concurrently in search of local optima. In addition, a particle swarm optimization rule is used to reinitialize neuronal states iteratively to escape from local minima toward better ones. Experimental results on eight benchmark datasets are elaborated to demonstrate the superior performance of the proposed approach against six baseline algorithms in terms of factorization error. Additionally, the viability of the proposed approach is demonstrated for pattern discovery on three datasets.

Dynamic optimization on quantum hardware: Feasibility for a process industry use case

The quest for real-time dynamic optimization solutions in the process industry represents a formidable computational challenge, particularly within the realm of applications like model-predictive control, where rapid and reliable computations are critical. Conventional methods can struggle to surmount the complexities of such tasks. Quantum computing and quantum annealing emerge as avant-garde contenders to transcend conventional computational constraints. We convert a dynamic optimization problem, characterized by an optimization problem with a system of differential–algebraic equations embedded, into a Quadratic Unconstrained Binary Optimization problem, enabling quantum computational approaches. The empirical findings synthesized from classical methods, simulated annealing, quantum annealing via D-Wave’s quantum annealer, and hybrid solver methodologies, illuminate the intricate landscape of computational prowess essential for tackling complex and high-dimensional dynamic optimization problems. Our findings suggest that while quantum annealing is a maturing technology that currently does not outperform state-of-the-art classical solvers, continuous improvements could eventually aid in increasing efficiency within the chemical process industry.

Multi-Objective Portfolio Optimization Using a Quantum Annealer

In this study, the portfolio optimization problem is explored, using a combination of classical and quantum computing techniques. The portfolio optimization problem with specific objectives or constraints is often a quadratic optimization problem, due to the quadratic nature of, for example, risk measures. Quantum computing is a promising solution for quadratic optimization problems, as it can leverage quantum annealing and quantum approximate optimization algorithms, which are expected to tackle these problems more efficiently. Quantum computing takes advantage of quantum phenomena like superposition and entanglement. In this paper, a specific problem is introduced, where a portfolio of loans need to be optimized for 2030, considering ‘Return on Capital’ and ‘Concentration Risk’ objectives, as well as a carbon footprint constraint. This paper introduces the formulation of the problem and how it can be optimized using quantum computing, using a reformulation of the problem as a quadratic unconstrained binary optimization (QUBO) problem. Two QUBO formulations are presented, each addressing different aspects of the problem. The QUBO formulation succeeded in finding solutions that met the emission constraint, although classical simulated annealing still outperformed quantum annealing in solving this QUBO, in terms of solutions close to the Pareto frontier. Overall, this paper provides insights into how quantum computing can address complex optimization problems in the financial sector. It also highlights the potential of quantum computing for providing more efficient and robust solutions for portfolio management.

Qubit Efficient Quantum Algorithms for the Vehicle Routing Problem on Noisy Intermediate‐Scale Quantum Processors

AbstractThe vehicle routing problem with time windows (VRPTW) is a common optimization problem faced within the logistics industry. In this work, the use of a previously‐introduced qubit encoding scheme is explored to reduce the number of qubits, to evaluate the effectiveness of Noisy Intermediate‐Scale Quantum (NISQ) devices when applied to industry relevant optimization problems. A quantum variational approach is applied to a testbed of multiple VRPTW instances ranging from 11 to 3964 routes. These intances are formulated as quadratic unconstrained binary optimization (QUBO) problems based on realistic shipping scenarios. The results are compared with standard binary‐to‐qubit mappings after executing on simulators as well as various quantum hardware platforms, including IBMQ, AWS (Rigetti), and IonQ. These results are benchmarked against the classical solver, Gurobi. The approach can find approximate solutions to the VRPTW comparable to those obtained from quantum algorithms using the full encoding, despite the reduction in qubits required. These results suggest that using the encoding scheme to fit larger problem sizes into fewer qubits is a promising step in using NISQ devices to find approximate solutions for industry‐based optimization problems, although additional resources are still required to eke out the performance from larger problem sizes.

Transformation of Elliptic Curve Discrete Logarithm Problem to QUBO Using Direct Method in Quantum Annealing Applications

This paper investigates how to reduce the elliptic curve discrete logarithm problem over prime fields to the quadratic unconstrained binary optimization (QUBO) problem in order to obtain as few logical qubits as possible. In the best case scenario, if n is the bitlength of a characteristic of prime field Fp, approximately 3n³ logical qubits are required for such a reduction in the Edwards curve case. We present a practical attack on an elliptic curve discrete logarithm problem over the 3-bit prime field F7 for an elliptic curve with the subgroup of order 8. We solved this problem using the D-Wave Advantage QPU. To the best of the authors' knowledge, no one has made, so far, a practical attack on the elliptic curve discrete logarithm over a prime field using the direct quantum method.

Quantum annealing-based computed tomography using variational approach for a real-number image reconstruction

Objective. Despite recent advancements in quantum computing, the limited number of available qubits has hindered progress in CT reconstruction. This study investigates the feasibility of utilizing quantum annealing-based computed tomography (QACT) with current quantum bit levels. Approach. The QACT algorithm aims to precisely solve quadratic unconstrained binary optimization problems. Furthermore, a novel approach is proposed to reconstruct images by approximating real numbers using the variational method. This approach allows for accurate CT image reconstruction using a small number of qubits. The study examines the impact of projection data quantity and noise on various image sizes ranging from 4 × 4 to 24 × 24 pixels. The reconstructed results are compared against conventional reconstruction algorithms, namely maximum likelihood expectation maximization (MLEM) and filtered back projection (FBP). Main result. By employing the variational approach and utilizing two qubits for each pixel of the image, accurate reconstruction was achieved with an adequate number of projections. Under conditions of abundant projections and lower noise levels, the image quality in QACT algorithm outperformed that of MLEM and FBP algorithms. However, in situations with limited projection data and in the presence of noise, the image quality in QACT was inferior to that in MLEM. Significance. This study developed the QACT reconstruction algorithm using the variational approach for real-number reconstruction. Remarkably, only 2 qubits were required for each pixel representation, demonstrating their sufficiency for accurate reconstruction.

Quantum computer-aided job scheduling for storage and retrieval systems

Abstract In this paper, a quantum computer-aided approach to job scheduling for automated storage and retrieval systems is introduced. The approach covers application cases, where various objects need to be transported between storage positions and the order of transport operations can be freely chosen. The objective of job scheduling is to arrange the transport operations in a sequence, where the cumulative costs of the transport operations and empty runs between subsequent transport operations are minimized. The scheduling problem is formulated as an asymmetric quadratic unconstrained binary optimization (QUBO) problem, in which the transport operations are modeled as nodes and empty runs are modeled as edges, with costs assigned to each node and each edge. An Quantum Approximate Optimization Algorithm (QAOA) is used to solve the QUBO. Evaluations of the quantum computer-aided job scheduling approach have been conducted on the IBM Q System One quantum computer in Ehningen. In particular, the running time for the solution of the QUBO has been investigated, as well as the scalability of the approach with respect to the required number of qubits.

Exploration of new chemical materials using black-box optimization with the D-wave quantum annealer

In materials informatics, searching for chemical materials with desired properties is challenging due to the vastness of the chemical space. Moreover, the high cost of evaluating properties necessitates a search with a few clues. In practice, there is also a demand for proposing compositions that are easily synthesizable. In the real world, such as in the exploration of chemical materials, it is common to encounter problems targeting black-box objective functions where formalizing the objective function in explicit form is challenging, and the evaluation cost is high. In recent research, a Bayesian optimization method has been proposed to formulate the quadratic unconstrained binary optimization (QUBO) problem as a surrogate model for black-box objective functions with discrete variables. Regarding this method, studies have been conducted using the D-Wave quantum annealer to optimize the acquisition function, which is based on the surrogate model and determines the next exploration point for the black-box objective function. In this paper, we address optimizing a black-box objective function containing discrete variables in the context of actual chemical material exploration. In this optimization problem, we demonstrate results obtaining parameters of the acquisition function by sampling from a probability distribution with variance can explore the solution space more extensively than in the case of no variance. As a result, we found combinations of substituents in compositions with the desired properties, which could only be discovered when we set an appropriate variance.

Individual subject evaluated difficulty of adjustable mazes generated using quantum annealing

In this study, the maze generation using quantum annealing is proposed. We reformulate a standard algorithm to generate a maze into a specific form of a quadratic unconstrained binary optimization problem suitable for the input of the quantum annealer. To generate more difficulty mazes, we introduce an additional cost function Qupdate to increase the difficulty. The difficulty of the mazes was evaluated by the time to solve the maze of 12 human subjects. To check the efficiency of our scheme to create the maze, we investigated the time-to-solution of a quantum processing unit, classical computer, and hybrid solver. The results show that Qupdate generates difficult mazes tailored to the individual. Furthermore, it show that the quantum processing unit is more efficient at generating mazes than other solvers. Finally, we also present applications how our results could be used in the future.

Cutting-Edge Trajectory Optimization through Quantum Annealing

This paper introduces an innovative approach to explore the capabilities of Quantum Annealing (QA) for trajectory optimization in dynamic systems. The proposed method involves transforming trajectory optimization problems into equivalent binary optimization problems using Quadratic Unconstrained Binary Optimization (QUBO) representation. The procedure is general and adaptable, making it applicable to a wide range of optimal control problems that entail the satisfaction of dynamic, boundary, and path constraints. Specifically, the trajectory is discretized and approximated using polynomials. In contrast to the conventional approach of determining the polynomial degree (n) solely based on the number of boundary conditions, a specific factor is introduced in our method to augment the polynomial degree. As a result, the ultimate polynomial degree is calculated as a composite of two components: n = l + (m−1), where m denotes the count of boundary conditions and l signifies the number of independent variables. By leveraging inverse dynamics, the control required to follow the approximated trajectory can be determined as a linear function of independent variables l. As a result, the optimization function, which is represented by the integral of the square of the control, can be formulated as a QUBO problem and the QA is employed to find the optimal binary solutions.

Bit duplication technique to generate hard quadratic unconstrained binary optimization problems with adjustable sizes

SummaryQuadratic unconstrained binary optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The main contribution of this article is to propose the bit duplication technique that can specify the number of duplicated bits, so that it can generate hard QUBO problem with adjustable sizes. The idea is to duplicate specified number of bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem with bits is converted to a hard QUBO problem with bits . We use random QUBO problems, N‐Queen problems, traveling salesman problem and maximum weight matching problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D‐Wave samplers with SA, D‐Wave hybrid and ABS2 QUBO solver are evaluated for solving these QUBO problems. The experimental results show that only a small scale of duplicated bits can make QUBO problems harder. Hence, the bit duplication technique is a potent method to generate hard QUBO problems and generated QUBO problems can be used as benchmark problems for evaluating the search performance of QUBO solvers.

Coordinated post-disaster restoration for resilient urban distribution systems: A hybrid quantum-classical approach

Incorporating multiple resilient resources into coordinated post-disaster restoration (CPR) strategy contributes positively to resilience enhancement of power distribution systems (PDS). However, the consideration of multiple factors may exacerbate model complexity, resulting in longer solution times and compromising the strategy’s practicality. With the remarkable potential power of quantum computing (QC), we propose a hybrid quantum-classical (HQC) approach for this problem. A HQC-based framework is proposed and the CPR model is established considering repair crew dispatch, PDS operation, microgrids operation, and topology reconfiguration. Then, a hybrid quantum-classical Benders decomposition (HQC-BD) algorithm and detailed pseudocode are proposed and presented in compact form. The integer slack and binary expansion methods are applied to transform the constrained problem into quadratic unconstrained binary optimization problem. Finally, the effectiveness and computational efficiency of the proposed HQC-BD approach are verified. The comparative analysis of different calculation methods is conducted by employing the D-Wave’s direct quantum processing unit solvers and hybrid solvers via the Leap™ quantum cloud service. This paper explores the possibility of quantum-accelerated resilience restoration and enhancement from the HQC point of view.

Quantum Annealing with Inequality Constraints: The Set Cover Problem

AbstractQuantum annealing is a promising method for solving hard optimization problems by transforming them into quadratic unconstrained binary optimization (QUBO) problems. However, when constraints are involved, particularly multiple inequality constraints, incorporating them into the objective function poses challenges. In this paper, the authors present two novel approaches for solving problems with multiple inequality constraints on a quantum annealer and apply them to the set cover problem (SCP). The first approach uses the augmented Lagrangian method to represent the constraints, while the second approach employs a higher‐order binary optimization (HUBO) formulation. The experiments show that both approaches outperform the standard approach for solving the SCP on the D‐Wave Advantage quantum annealer. The HUBO formulation performs slightly better than the augmented Lagrangian method in solving the SCP, but its scalability in terms of embeddability in the quantum chip is worse. The results demonstrate that the proposed augmented Lagrangian and HUBO methods can successfully implement a large number of inequality constraints, making them applicable to a broad range of constrained problems beyond the SCP.

Universal Quantum Optimization with Cold Atoms in an Optical Cavity.

Cold atoms in an optical cavity have been widely used for quantum simulations of many-body physics, where the quantum control capability has been advancing rapidly in recent years. Here, we show the atom cavity system is universal for quantum optimization with arbitrary connectivity. We consider a single-mode cavity and develop a Raman coupling scheme by which the engineered quantum Hamiltonian for atoms directly encodes number partition problems. The programmability is introduced by placing the atoms at different positions in the cavity with optical tweezers. The number partition problem solution is encoded in the ground state of atomic qubits coupled through a photonic cavity mode, which can be reached by adiabatic quantum computing. We construct an explicit mapping for the 3-SAT and vertex cover problems to be efficiently encoded by the cavity system, which costs linear overhead in the number of atomic qubits. The atom cavity encoding is further extended to quadratic unconstrained binary optimization problems. The encoding protocol is optimal in the cost of atom number scaling with the number of binary degrees of freedom of the computation problem. Our theory implies the atom cavity system is a promising quantum optimization platform searching for practical quantum advantage.

An Overview of Algorithms for Solving Vehicle Routing Problems in the Quantum-Classical Cloud

Introduction. The hope of solving the problem of the avalanche-like growth of requirements for computing power, essential for solving complex routing problems and other problems of combinatorial optimization, relies on the latest quantum computers, in the development of which governments and corporations invest multi-billion investments. The article examines modern routing algorithms and performs their analysis and verification, if the authors of the algorithm provided appropriate test programs. The purpose of the article is to review the current state of development in the field of development of routing algorithms for hybrid quantum-classical clouds, analyze them and propose a classification of algorithms. Results. Modern quantum computers (QCs) make it possible to find approximate solutions to some of mathematical problems faster than classical computers. The inaccuracy of the solutions obtained by the QC is a consequence of physical and technological limitations: calculation errors are caused by thermal noise, a small number of computational elements - qubits and connections between them, which requires the decomposition of the problem and the use of heuristic algorithms. The analysis of approaches to the solution of optimization problems on QC allows us to single out: quantum response and variational search of eigenvalues based on quantum logic gates as the general directions of development of the vast majority of algorithms for solving routing problems. The considered algorithms reduce the vehicle routing problem to a quadratic unconstrained binary optimization problem, which is isomorphic to the Hamilton-Ising model. In this form, the problem is suitable for embedding in QC, which finds an approximate solution that has the best statistical reliability or corresponds to the quantum state with the lowest energy. As a separate class, vehicle routing algorithms for classical computers that use quantum computing to accelerate problem solving can be distinguished. For example, neural networks that calculate weighting factors using QC or an ant algorithm that calculates a pheromone trail in a hybrid cloud. It should be mentioned the quantum-inspired algorithms, which are based on software tools for the simulation of QC and the corresponding libraries and allow creating an effective class of algorithms for solving problems of vehicle routing. Conclusions. Combining hardware quantum annealing with a number of software tools for calculating optimization problems for classical computers in a hybrid quantum-classical cloud service allows to obtain advantages in speed and accuracy of some types of complex optimization problems of a commercial scale, in particular, routing vehicles, which is already bringing substantial profits to a number of corporations. Keywords: vehicle routing problem, quantum computer, annealing, combinatorial optimization, traveling salesman problem, clustering, qubit.

Quantum computing for transport network design problems

Transport network design problem (TNDP) is a well-studied problem for planning and operations of transportation systems. They are widely used to determine links for capacity enhancement, link closures to schedule maintenance, identify new road or transit links and more generally network enhancements under resource constraints. As changes in network capacities result in a redistribution of demand on the network, resulting in changes in the congestion patterns, TNDP is generally modelled as a bi-level problem, which is known to be NP-hard. Meta-heuristic methods, such as Tabu Search Method are relied upon to solve these problems, which have been demonstrated to achieve near optimality in reasonable time. The advent of quantum computing has afforded an opportunity to solve these problems faster. We formulate the TNDP problem as a bi-level problem, with the upper level formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem that is solved using quantum annealing on a D-Wave quantum computer. We compare the results with Tabu Search. We find that quantum annealing provides significant computational benefit. The proposed solution has implications for networks across different contexts including communications, traffic, industrial operations, electricity, water, broader supply chains and epidemiology.

Network Bandwidth Utilization Optimization with Quantum-Inspired Approach

As the popularity of mobile devices and services continues to increase, the demand for more efficient and effective algorithms to manage network utility is increased. Here, we aim to optimize Network Bandwidth Utilization (NBU) by managing user equipments (UEs) and base stations (BSs) association in 5G ORAN architecture. We consider the network total throughput as an objective function and try to find the suitable Cell Individual Offset (CIO) to trigger UE handover with the A3 event such that the network throughput is optimal. We formulate this problem into a Quadratic Unconstrained Binary Optimization (QUBO) problem such that we can explore the power of Quantum Annealing (QA), which is a promising approach to finding the optimal solution for the NP-Hard problem. We also analyze the penalty coefficients setting problem which impacts the outcome of QA significantly. We test our QUBO formulation on a dataset of 7 BSs and 150 UEs with Fujitsu’s third generation Digital Annealer (DA3). Using our QUBO formulation with DA3 solver with fine-tuning, we can improve the throughput with 60% hit rate on this dataset.

Quantum computing for a profusion of postman problem variants

In this paper we study the viability of solving the Chinese Postman Problem, a graph routing optimization problem, and many of its variants on a quantum annealing device. Routing problem variants considered include graph type, directionally varying weights, number of parties involved in routing, among others. We put emphasis on the explanation of how to convert such problems into quadratic unconstrained binary optimization (QUBO) problems. QUBO is one of two equivalent natural paradigms for quantum annealing devices, the other being the Ising Model. We also expand upon a previously discovered algorithm for solving the Chinese Postman Problem on a closed undirected graph to decrease the number of constraints and variables used in the problem. Optimal annealing parameter settings and constraint weight values are discussed based on results from implementation on the D-Wave 2000Q and Advantage. Results from classical, purely quantum, and hybrid algorithms are compared.