Quadratic Unconstrained Binary Optimization Research Articles

Reduced-order modeling on a near-term quantum computer

Quantum computing is an advancing area of research in which computer hardware and algorithms are developed to take advantage of quantum mechanical phenomena. In recent studies, quantum algorithms have shown promise in solving linear systems of equations as well as systems of linear ordinary differential equations (ODEs) and partial differential equations (PDEs). Reduced-order modeling (ROM) algorithms for studying fluid dynamics have shown success in identifying linear operators that can describe flowfields, where dynamic mode decomposition (DMD) is a particularly useful method in which a linear operator is identified from data. In this work, DMD is reformulated as an optimization problem to propagate the state of the linearized dynamical system on a quantum computer. This reformulation was chosen as a means of facilitating implementation on a near-term quantum computer. Quadratic unconstrained binary optimization (QUBO), a technique for optimizing quadratic polynomials in binary variables, allows for quantum annealing algorithms to be applied. A quantum circuit model (quantum approximation optimization algorithm, QAOA) is utilized to obtain predictions of the state trajectories. Results are shown for the quantum-ROM predictions for flow over a 2D cylinder at Re = 220 and flow over a NACA0009 airfoil at Re = 500 and α=15∘. The quantum-ROM predictions are found to depend on the number of bits utilized for a fixed point representation and the truncation level of the DMD model. Comparisons with DMD predictions from a classical computer algorithm are made, as well as an analysis of the computational complexity and prospects for future, more fault-tolerant quantum computers.

Quantum-enhanced hybrid feature engineering in thoracic CT image analysis for state-of-the-art nodule classification: an advanced lung cancer assessment

The intricate nature of lung cancer treatment poses considerable challenges upon diagnosis. Early detection plays a pivotal role in mitigating its escalating global mortality rates. Consequently, there are pressing demands for robust and dependable early detection and diagnostic systems. However, the technological limitations and complexity of the disease make it challenging to implement an efficient lung cancer screening system. AI-based CT image analysis techniques are showing significant contributions to the development of computer-assisted detection (CAD) systems for lung cancer screening. Various existing research groups are working on implementing CT image analysis systems for assessing and classifying lung cancer. However, the complexity of different structures inside the CT image is high and comprehension of significant information inherited by them is more complex even after applying advanced feature extraction and feature selection techniques. Traditional and classical feature selection techniques may struggle to capture complex interdependencies between features. They may get stuck in local optima and sometimes require additional exploration strategies. Traditional techniques may also struggle with combinatorial optimization problems when applied to a prominent feature space. This paper proposed a methodology to overcome the existing challenges by applying feature extraction using Vision Transformer (FexViT) and Feature selection using the Quantum Computing based Quadratic unconstrained binary optimization (QC-FSelQUBO) technique. This algorithm shows better performance when compared with other existing techniques. The proposed methodology showed better performance as compared to other existing techniques when evaluated by applying necessary output measures, such as accuracy, Area under roc (receiver operating characteristics) curve, precision, sensitivity, and specificity, obtained as 94.28%, 99.10%, 96.17%, 90.16% and 97.46%. The further advancement of CAD systems is essential to meet the demand for more reliable detection and diagnosis of cancer, which can be addressed by leading the proposed quantum computation and growing AI-based technology ahead.

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.

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.

Partitioning QUBO with two-way one-hot conditions on traveling salesman problems for city distributions with multiple clusters

We introduce a method for solving a quadratic unconstrained binary optimization (QUBO) with the two-way one-hot constraints by dividing the QUBO into parts and solving it with an Ising machine. The one-hot constraint is a constraint condition in that only one binary variable takes the value one for a set of multiple binary variables. The two-way one-hot constraint imposes one-hot constraint conditions on every row and every column of a two-dimensional array of binary variables with equal numbers of rows and columns. For example, traveling salesman problems (TSPs) have two-way one-hot constraints. We propose two methods to solve a TSP by dividing the cities into clusters based on information in the QUBO matrix. The proposed methods decompose cities into clusters and solve TSPs for each cluster. We solve TSPs such that the cities are distributed in clusters and compare the results of the two proposed methods with the results of solving the problem without partitioning. The results show that the proposed methods are robust to the coefficient parameters in the Hamiltonian of QUBO and can obtain a solution closer to the optimal solution when solving the problem without partitioning is hard. We also discuss the application of the methods proposed in the TSPs to the quadratic assignment problems and to the problems of ordering things.

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.

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.

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.

Unbalanced penalization: a new approach to encode inequality constraints of combinatorial problems for quantum optimization algorithms

Solving combinatorial optimization problems of the kind that can be codified by quadratic unconstrained binary optimization (QUBO) is a promising application of quantum computation. Some problems of this class suitable for practical applications such as the traveling salesman problem (TSP), the bin packing problem (BPP), or the knapsack problem (KP) have inequality constraints that require a particular cost function encoding. The common approach is the use of slack variables to represent the inequality constraints in the cost function. However, the use of slack variables considerably increases the number of qubits and operations required to solve these problems using quantum devices. In this work, we present an alternative method that does not require extra slack variables and consists of using an unbalanced penalization function to represent the inequality constraints in the QUBO. This function is characterized by larger penalization when the inequality constraint is not achieved than when it is. We evaluate our approach on the TSP, BPP, and KP, successfully encoding the optimal solution of the original optimization problem near the ground state cost Hamiltonian. Additionally, we employ D-Wave Advantage and D-Wave hybrid solvers to solve the BPP, surpassing the performance of the slack variables approach by achieving solutions for up to 29 items, whereas the slack variables approach only handles up to 11 items. This new approach can be used to solve combinatorial problems with inequality constraints with a reduced number of resources compared to the slack variables approach using quantum annealing or variational quantum algorithms.

Ferroelectric compute-in-memory annealer for combinatorial optimization problems

Computationally hard combinatorial optimization problems (COPs) are ubiquitous in many applications. Various digital annealers, dynamical Ising machines, and quantum/photonic systems have been developed for solving COPs, but they still suffer from the memory access issue, scalability, restricted applicability to certain types of COPs, and VLSI-incompatibility, respectively. Here we report a ferroelectric field effect transistor (FeFET) based compute-in-memory (CiM) annealer for solving larger-scale COPs efficiently. Our CiM annealer converts COPs into quadratic unconstrained binary optimization (QUBO) formulations, and uniquely accelerates in-situ the core vector-matrix-vector (VMV) multiplication operations of QUBO formulations in a single step. Specifically, the three-terminal FeFET structure allows for lossless compression of the stored QUBO matrix, achieving a remarkably 75% chip size saving when solving Max-Cut problems. A multi-epoch simulated annealing (MESA) algorithm is proposed for efficient annealing, achieving up to 27% better solution and ~ 2X speedup than conventional simulated annealing. Experimental validation is performed using the first integrated FeFET chip on 28nm HKMG CMOS technology, indicating great promise of FeFET CiM array in solving general COPs.

QAL-BP: an augmented Lagrangian quantum approach for bin packing

The bin packing is a well-known NP-Hard problem in the domain of artificial intelligence, posing significant challenges in finding efficient solutions. Conversely, recent advancements in quantum technologies have shown promising potential for achieving substantial computational speedup, particularly in certain problem classes, such as combinatorial optimization. In this study, we introduce QAL-BP, a novel Quadratic Unconstrained Binary Optimization (QUBO) formulation designed specifically for bin packing and suitable for quantum computation. QAL-BP utilizes the Augmented Lagrangian method to incorporate the bin packing constraints into the objective function while also facilitating an analytical estimation of heuristic, but empirically robust, penalty multipliers. This approach leads to a more versatile and generalizable model that eliminates the need for empirically calculating instance-dependent Lagrangian coefficients, a requirement commonly encountered in alternative QUBO formulations for similar problems. To assess the effectiveness of our proposed approach, we conduct experiments on a set of bin packing instances using a real Quantum Annealing device. Additionally, we compare the results with those obtained from two different classical solvers, namely simulated annealing and Gurobi. The experimental findings not only confirm the correctness of the proposed formulation, but also demonstrate the potential of quantum computation in effectively solving the bin packing problem, particularly as more reliable quantum technology becomes available.

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.

HUBO & QUBO and Prime Factorization

This document details the methodology and steps taken to convert Higher Order Unconstrained Binary Optimization (HUBO) models into Quadratic Unconstrained Binary Optimization (QUBO) models. The focus is primarily on prime factorization problems; a critical and computationally intensive task relevant in various domains including cryptography, optimization, and number theory. The conversion from Higher-Order Binary Optimization (HUBO) to Quadratic Unconstrained Binary Optimization (QUBO) models is crucial for harnessing the capabilities of advanced computing methodologies, particularly quantum computing and DYNEX neuromorphic computing. Quantum computing offers potential exponential speedups for specific problems through its intrinsic parallelism capabilities. Conversely, DYNEX neuromorphic computing enhances efficiency and accelerates the resolution of intricate, pattern-oriented tasks by simulating memristors in GPUs, employing a highly decentralized approach, via Blockchain technology. This transformation enables the exploitation of these cutting-edge computing paradigms to address complex optimization challenges effectively. Through detailed explanations, mathematical formulations, and algorithmic strategies, this document aims to provide a comprehensive guide to understanding and implementing the conversion process from HUBO to QUBO. It underscores the importance of such transformations in making prime factorization computationally feasible on both existing classical computers and emerging computing technologies.

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 and quantum-inspired optimization for an in-core fuel management problem

Operation management of nuclear power plants consists of several computationally hard problems. Searching for an in-core fuel loading pattern is among them. The main challenge of this combinatorial optimization problem is the exponential growth of the search space with a number of loading elements. Here we study a reloading problem in a Quadratic Unconstrained Binary Optimization (QUBO) form. Such a form allows us to apply various techniques, including quantum annealing, classical simulated annealing, and quantum-inspired algorithm in order to find fuel reloading patterns for several realistic configurations of nuclear reactors. We present the results of benchmarking the in-core fuel management problem in the QUBO form using the aforementioned computational techniques. This work demonstrates potential applications of quantum computers and quantum-inspired algorithms in the energy industry.

Quantum and quantum-inspired optimization for solving the minimum bin packing problem

Quantum computing devices are believed to be powerful in solving hard computational tasks, in particular, combinatorial optimization problems. In the present work, we consider a particular type of the minimum bin packing problem, which can be used for solving the problem of filling spent nuclear fuel in deep-repository canisters that is relevant for atomic energy industry. We first redefine the aforementioned problem it in terms of quadratic unconstrained binary optimization. Such a representation is natively compatible with existing quantum annealing devices as well as quantum-inspired algorithms. We then present the results of the numerical comparison of quantum and quantum-inspired methods. Results of our study indicate on the possibility to solve industry-relevant problems of atomic energy industry using quantum and quantum-inspired optimization.

Quadratic Unconstrained Binary Optimization Approach for Incorporating Solvency Capital into Portfolio Optimization

In this paper, we consider the inclusion of the solvency capital requirement (SCR) into portfolio optimization by the use of a quadratic proxy model. The Solvency II directive requires insurance companies to calculate their SCR based on the complete loss distribution for the upcoming year. Since this task is, in general, computationally challenging for insurance companies (and therefore, not taken into account during portfolio optimization), employing more feasible proxy models provides a potential solution to this computational difficulty. Here, we present an approach that is also suitable for future applications in quantum computing. We analyze the approximability of the solvency capital ratio in a quadratic form using machine learning techniques. This allows for an easier consideration of the SCR in the classical mean-variance analysis. In addition, it allows the problem to be formulated as a quadratic unconstrained binary optimization (QUBO), which benefits from the potential speedup of quantum computing. We provide a detailed description of our model and the translation into a QUBO. Furthermore, we investigate the performance of our approach through experimental studies.

Multi-satellite cooperative scheduling method for large-scale tasks based on hybrid graph neural network and metaheuristic algorithm

Traditional heuristic optimization algorithms are no longer applicable to the multiple satellites scheduling with large-scale tasks, as they are unable to provide satisfactory performance in terms of convergence and speed when addressing complex constraint conflicts. Therefore, we propose a multi-satellite cooperative scheduling method for large-scale tasks based on a hybrid graph neural network (GNN) and metaheuristic algorithm. By using the representation and extraction capability of GNN for relations in graph, the features of large-scale tasks and their constraint relations were expressed, and the generalized knowledge was extracted. And combined with metaheuristic algorithm, the optimization framework for large-scale satellite task scheduling based knowledge and data was implemented. The method consists of a GNN pre-solver module for static constraint conflicts and a metaheuristic optimization module for dynamic constraint conflicts of satellite missions. In the GNN pre-solver module, the graph sample and aggregate network was used to learn and extract the common knowledge in the static task-conflict graph, and provided effective prior information for the metaheuristic optimization module. The quadratic unconstrained binary optimization loss function was designed with multiple influencing factors, to convert the discrete optimization function into a continuous function; a greedy threshold was also added to improve the task completion rate. Finally, numerical experimental results showed that the proposed method can achieve an efficient solution to the multi-satellite scheduling problem with tens of thousands tasks. Compared with the commonly used multi-satellite scheduling algorithms (the genetic algorithm (GA) and greedy-GA), the proposed method can obtain higher-quality solutions under the same conditions and greatly improve the computational efficiency of large-scale mission planning.

D-Wave Implementation of Quantum Annealing for Optimal Resource Allocation in Disaster Response Operation of Marikina City

Quantum computing shows a positive approach for addressing optimization challenges in NP-hard problems such as the vehicle routing problem (VRP). This study focuses on improving the efficiency of disaster response operations by localizing the application of D-wave quantum annealing in Marikina City. This study uses the Solution Partitioning Solver (SPS) and the Quadratic Unconstrained Binary Optimization (QUBO) formulation to convert the VRP into an equation that can be solved using quantum annealing. The study demonstrates that quantum computing effectively distributes resources during emergency response operations and improves overall operational efficiency. In determining the most effective route for each vehicle, the D-wave Leap API and QUBO representation compute the distances traveled by each vehicle. These findings contribute to the practical applications of quantum computing to revolutionize various fields, including disaster management. Implementing D-wave quantum annealing in Marikina City shows relevance for future advancements in optimizing resource allocation and improving disaster response operations.

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.