Quadratic Unconstrained Binary Optimization Research Articles

Harnessing quantum power: Revolutionizing materials design through advanced quantum computation

AbstractThe design of advanced materials for applications in areas of photovoltaics, energy storage, and structural engineering has made significant strides. However, the rapid proliferation of candidate materials—characterized by structural complexity that complicates the relationships between features—presents substantial challenges in manufacturing, fabrication, and characterization. This review introduces a comprehensive methodology for materials design using cutting‐edge quantum computing, with a particular focus on quadratic unconstrained binary optimization (QUBO) and quantum machine learning (QML). We introduce the loop framework for QUBO‐empowered materials design, including constructing high‐quality datasets that capture critical material properties, employing tailored computational methods for precise material modeling, developing advanced figures of merit to evaluate performance metrics, and utilizing quantum optimization algorithms to discover optimal materials. In addition, we delve into the core principles of QML and illustrate its transformative potential in accelerating material discovery through a range of quantum simulations and innovative adaptations. The review also highlights advanced active learning strategies that integrate quantum artificial intelligence, offering a more efficient pathway to explore the vast, complex material design space. Finally, we discuss the key challenges and future opportunities for QML in material design, emphasizing their potential to revolutionize the field and facilitate groundbreaking innovations.

Atom cavity encoding for NP-complete problems

We consider an atom-cavity system having long-range atomic interactions mediated by cavity modes. It has been shown that quantum simulations of spin models with this system can naturally be used to solve number partition problems. Here, we present encoding schemes for numerous nondeterministic polynomial-time complete (NP-complete) problems, encompassing the majority of Karp’s 21 NP-complete problems. We find a number of such computation problems can be encoded by the atom-cavity system at a linear cost of atom number. There are still certain problems that cannot be encoded by the atom-cavity as efficiently, such as quadratic unconstrained binary optimization (QUBO), and the Hamiltonian cycle. For these problems, we provide encoding schemes with a quadratic or quartic cost in the atom number. We expect this work to provide important guidance to search for the practical quantum advantage of the atom-cavity system in solving NP-complete problems. Moreover, the encoding schemes we develop here may also be adopted in other optical systems for solving NP-complete problems, where a similar form of Mattis-type spin glass Hamiltonian as in the atom-cavity system can be implemented.

Quantum Computing in Community Detection for Anti-Fraud Applications

Fraud detection within transaction data is crucial for maintaining financial security, especially in the era of big data. This paper introduces a novel fraud detection method that utilizes quantum computing to implement community detection in transaction networks. We model transaction data as an undirected graph, where nodes represent accounts and edges indicate transactions between them. A modularity function is defined to measure the community structure of the graph. By optimizing this function through the Quadratic Unconstrained Binary Optimization (QUBO) model, we identify the optimal community structure, which is then used to assess the fraud risk within each community. Using a Coherent Ising Machine (CIM) to solve the QUBO model, we successfully divide 308 nodes into four communities. We find that the CIM computes faster than the classical Louvain and simulated annealing (SA) algorithms. Moreover, the CIM achieves better community structure than Louvain and SA as quantified by the modularity function. The structure also unambiguously identifies a high-risk community, which contains almost 70% of all the fraudulent accounts, demonstrating the practical utility of the method for banks’ anti-fraud business.

Benchmarking the optimization of optical machines with the planted solutions

This research focuses on developing effective benchmarks for quadratic unconstrained binary optimization instances, crucial for evaluating the performance of Ising hardware and solvers. Currently, the field lacks accessible and reproducible models for systematically testing such systems, particularly in terms of detailed phase space characterization. Here, we introduce universal generative models based on an extension of Hebb’s rule of associative memory with asymmetric pattern weights. We conduct comprehensive calculations across different scales and dynamical equations, examining outcomes like the probabilities of reaching the ground state, planted state, spurious state, or other energy levels. Additionally, the generated problems reveal properties such as the easy-hard-easy complexity transition and complex solution cluster structures. This method offers a promising platform for analyzing and understanding the behavior of physical hardware and its simulations, contributing to future advancements in optimization technologies.

16-channel photonic solver for optimization problems on a silicon chip

A programmable photonic solver for quadratic unconstrained binary optimization (QUBO) problems is demonstrated with a hybrid optoelectronic scheme, which consists of a photonic chip and an electronic driving board. The photonic chip is employed to perform the optical vector-matrix multiplication (OVMM) to calculate the cost function of the QUBO problem, while the electronic processor runs the heuristic algorithm to search for the optimal solution. Due to the parallel and low-latency propagation of lightwave, the calculation of the cost function can be accelerated. The photonic chip is fabricated on the silicon on insulator (SOI) substrate and integrates 16 high-speed electro-optic modulators, 88 thermo-optic phase shifters, and 16 balanced photodetectors. The computing speed of the photonic chip is 1.66 TFLOP/s. As a proof of principle, two randomly generated 16-dimensional QUBO problems are solved with high successful probabilities. These results present the potential of fast solving optimization problems with integrated photonic systems.

Novel Application of Quantum Computing for Routing and Spectrum Assignment in Flexi-Grid Optical Networks

Flexi-grid technology has revolutionized optical networking by enabling Elastic Optical Networks (EONs) that offer greater flexibility and dynamism compared to traditional fixed-grid systems. As data traffic continues to grow exponentially, the need for efficient and scalable solutions to the routing and spectrum assignment (RSA) problem in EONs becomes increasingly critical. The RSA problem, being NP-Hard, requires solutions that can simultaneously address both spatial routing and spectrum allocation. This paper proposes a novel quantum-based approach to solving the RSA problem. By formulating the problem as a Quadratic Unconstrained Binary Optimization (QUBO) model, we employ the Quantum Approximate Optimization Algorithm (QAOA) to effectively solve it. Our approach is specifically designed to minimize end-to-end delay while satisfying the continuity and contiguity constraints of frequency slots. Simulations conducted using the Qiskit framework and IBM-QASM simulator validate the effectiveness of our method. We applied the QAOA-based RSA approach to small network topology, where the number of nodes and frequency slots was constrained by the limited qubit count on current quantum simulator. In this small network, the algorithm successfully converged to an optimal solution in less than 30 iterations, with a total runtime of approximately 10.7 s with an accuracy of 78.8%. Additionally, we conducted a comparative analysis between QAOA, integer linear programming, and deep reinforcement learning methods to evaluate the performance of the quantum-based approach relative to classical techniques. This work lays the foundation for future exploration of quantum computing in solving large-scale RSA problems in EONs, with the prospect of achieving quantum advantage as quantum technology continues to advance.

Implementation of spectral methods on Ising machines: toward flow simulations on quantum annealers

Abstract We investigate the possibility and current limitations of flow computations using quantum annealers by solving a fundamental flow problem on Ising machines. As a fundamental problem, we consider the one-dimensional advection-diffusion equation. We formulate it in a form suited to Ising machines (i.e., both classical and quantum annealers), perform extensive numerical tests on a classical annealer, and finally test it on an actual quantum annealer. To make it possible to process with an Ising machine, the problem is formulated as a minimization problem of the residual of the governing equation discretized using either the spectral method or the finite difference method. The resulting system equation is then converted to the Quadratic Unconstrained Binary Optimization (QUBO) form through the quantization of variables. It is found in numerical tests using a classical annealer that the spectral method requiring a smaller number of variables has a particular merit over
the finite difference method because the accuracy deteriorates with the increase of the number of variables. We also found that the computational error varies depending on the condition number of the coefficient matrix. In addition, we extended it to a two-dimensional problem and confirmed its fundamental applicability. From the numerical test using a quantum annealer, however, it turns out that the computation using a quantum annealer is still challenging due largely to the structural difference from the classical annealer, which leaves a number of issues toward its practical use.

Optimal box contraction for solving linear systems via simulated and quantum annealing

Solving linear systems of equations is an important problem in engineering. Many quantum algorithms, such as the Harrow–Hassidim–Lloyd algorithm and the box algorithm, have been proposed for solving such systems. The focus of this article is on improving the efficiency of the box algorithm. The basic principle behind this algorithm is to transform the linear system into a series of quadratic unconstrained binary optimization (QUBO) problems, which are then solved on annealing machines. The computational efficiency of the box algorithm is entirely determined by the number of iterations, which, in turn, depends on the box contraction ratio, typically set to 0.5. Here, it is shown through theoretical analysis that a contraction ratio of 0.5 is sub-optimal and that a computational speed-up can be achieved with a contraction ratio of 0.2. This is confirmed through numerical experiments where a computational speed-up between 20 % to 60 % is observed when the optimal contraction ratio is used.

Power flow analysis using quantum and digital annealers: a discrete combinatorial optimization approach

Power flow (PF) analysis is a foundational computational method to study the flow of power in an electrical network. This analysis involves solving a set of non-linear and non-convex differential-algebraic equations. State-of-the-art solvers for PF analysis, therefore, face challenges with scalability and convergence, specifically for large-scale and/or ill-conditioned cases characterized by high penetration of renewable energy sources, among others. The adiabatic quantum computing paradigm has been proven to efficiently find solutions for combinatorial problems in the noisy intermediate-scale quantum (NISQ) era, and it can potentially address the limitations posed by state-of-the-art PF solvers. For the first time, we propose a novel adiabatic quantum computing approach for efficient PF analysis. Our key contributions are (i) a combinatorial PF algorithm and a modified version that aligns with the principles of PF analysis, termed the adiabatic quantum PF algorithm (AQPF), both of which use Quadratic Unconstrained Binary Optimization (QUBO) and Ising model formulations; (ii) a scalability study of the AQPF algorithm; and (iii) an extension of the AQPF algorithm to handle larger problem sizes using a partitioned approach. Numerical experiments are conducted using different test system sizes on D-Wave’s Advantage™ quantum annealer, Fujitsu’s digital annealer V3, D-Wave’s quantum-classical hybrid annealer, and two simulated annealers running on classical computer hardware. The reported results demonstrate the effectiveness and high accuracy of the proposed AQPF algorithm and its potential to speed up the PF analysis process while handling ill-conditioned cases using quantum and quantum-inspired algorithms.

Hybrid quantum annealing decomposition framework for unit commitment

Quantum computing is an emerging and promising technology that has overwhelming quantum advantages compared to its classical counterparts. Unit commitment (UC) is a critical issue in the power system, and it becomes more challenging with the integration of intermittent renewable energy. Therefore, this paper proposes an innovative decomposition and coordination optimization framework to accelerate the solution of UC, in which the interaction between an adiabatic quantum computer and a classical computer is designed to harness the immense computational power of quantum computers effectively. First, decomposition methods considering the requirements of quantum computers are introduced to decompose UC into small-scale models. Then, the paper presents a quadratic unconstrained binary optimization modeling method to transform UC problems into the form of quantum computing. Furthermore, due to the limitations of quantum computing resources, a reductive variable technique is proposed to reduce the number of slack variables in the optimization model and ensure that it remains feasible for quantum computers. Case studies conducted in test systems with a quantum annealing simulator and a real quantum annealing computer illustrate the feasibility and effectiveness of the method and demonstrate its potential in the era of quantum computing.

Support vector machine based on the quadratic unconstrained binary optimization model

Abstract Support vector machine (SVM) is a powerful supervised machine learning model that is often used in binary classification algorithms. As Moore’s Law approaches its theoretical limits and the demand for machine learning to handle large-scale, high-dimensional data analysis intensifies, the necessity of adopting non-traditional computational approaches becomes evident. Quantum computing, in particular, emerges as a vital solution for the effective training of SVM models, providing capabilities beyond those of classical computing systems. To solve the above problems, a QUBO (quadratic unconstrained binary optimization) model is proposed to transform the SVM machine learning model into a quadratic unconstrained binary optimization problem so that they can be effectively trained on the D-Wave platform using adiabatic quantum computer. The results show that the QUBO model can transform the SVM model into a simple quadratic programming problem, which makes it suitable for adiabatic quantum computer processing. When processing large-scale and high-dimensional data, this transformation shows a natural advantage and significantly improves computational efficiency. The application potential of this transformation technology is huge in the medical field.

Geometric Landscape Annealing as an Optimization Principle Underlying the Coherent Ising Machine

Given the fundamental importance of combinatorial optimization across many diverse domains, there has been widespread interest in the development of unconventional physical computing architectures that can deliver better solutions with lower resource costs. However, a theoretical understanding of their performance remains elusive. We develop such understanding for the case of the coherent Ising machine (CIM), a network of optical parametric oscillators that can be applied to any quadratic unconstrained binary optimization problem. We focus on how the CIM finds low-energy solutions of the Sherrington-Kirkpatrick spin glass. As the laser gain of this system is annealed, the CIM interpolates between gradient descent on coupled soft spins to descent on coupled binary spins. By combining the Kac-Rice formula, the replica method, and supersymmetry breaking, we develop a detailed understanding of the evolving geometry of the high-dimensional energy landscape of the CIM as the laser gain increases, finding several phase transitions in the landscape, from flat to rough to rigid. Additionally, we develop a novel cavity method that provides a geometric interpretation of supersymmetry breaking in terms of the reactivity of a rough landscape to specific external perturbations. Our energy landscape theory successfully matches numerical experiments, provides geometric insights into the principles of CIM operation, and yields optimal annealing schedules. Published by the American Physical Society 2024

Digital annealing optimization for natural product structure elucidation.

The digital annealer (DA) leverages its computational capabilities of up to 100000 bits to address the complex nondeterministic polynomial-time (NP)-complete challenge inherent in elucidating complex structures of natural products. Conventional computational methods often face limitations with complex mixtures, as they struggle to manage the high dimensionality and intertwined relationships typical in natural products, resulting in inefficiencies and inaccuracies. This study reformulates the challenge into a Quadratic Unconstrained Binary Optimization framework, thereby harnessing the quantum-inspired computing power of the DA. Utilizing mass spectrometry data from three distinct herb species and various potential scaffolds, the DA proficiently locates optimal sidechain combinations that adhere to predefined target molecular weights. This methodology enhances the probability of selecting appropriate sidechains and substituted positions and ensures the generation of solutions within a reasonable 5-min window. The findings underscore the transformative potential of the DA in the realms of analytical chemistry and drug discovery, markedly improving both the precision and practicality of natural product structure elucidation.

Hybrid Ising Machine and Reinforcement Learning Approach to Folding Lattice Proteins

The hydrophobic polarity (HP) protein folding model is a simplified computation model utilized to study the folding of proteins into their three-dimensional structures. Understanding how proteins fold and misfold is fundamental to designing effective, safe drugs, allowing for targeted therapies and better patient outcomes. The amino acids of a protein can be categorized as either hydrophobic or polar, where the hydrophobic beads tend to cluster together in a properly folded protein to minimize exposure to the surrounding environment. Determining the lowest energy configurations of the HP model is a non-deterministic polynomial (NP) hard problem. One method is formulating the HP model as a quadratic unconstrained binary optimization (QUBO) problem which can be mapped to an Ising machine. In the Ising model, the lowest energy state of the system represents the correct lattice configuration of the folded protein. The Ising machine technique utilized of simulated annealing allows for future application of the work on an integrated silicon photonic chip. Alternatively, reinforcement learning designed with an energy-based reward function can optimize the HP model by adapting to the environment without the consequence of remaining in local minima like that of the Ising machine. A hybrid pipeline combining the Ising machine and reinforcement learning sequentially was designed to determine the two-dimensional lattice configurations of various protein chain lengths and sequences, combining the parallel optimization of the Ising machine on an integrated photonic platform and the adaptability of the reinforcement learning algorithm. Results found that this hybrid model improved the accuracy and ability of determining the lowest energy state in contrast to the Ising machine alone as well as improved the efficiency by reducing the number of iterations required when compared solely to reinforcement learning. Future work focuses on implementing a parallel pipeline approach as well as adapting to a three-dimensional model.

Solving the homogeneous Bethe-Salpeter equation with a quantum annealer

The homogeneous Bethe-Salpeter equation (hBSE), describing a bound system in a genuinely relativistic quantum-field theory framework, was solved for the first time by using a D-Wave quantum annealer. After applying standard techniques of discretization, the hBSE, in ladder approximation, can be formally transformed in a generalized eigenvalue problem (GEVP), with two square matrices: one symmetric and the other nonsymmetric. The latter matrix poses the challenge of obtaining a suitable formal approach for investigating the GEVP by means of a quantum annealer, i.e., to recast it as a quadratic unconstrained binary optimization problem. A broad numerical analysis of the proposed algorithms, applied to matrices of dimension up to 64, was carried out by using both the simulated-annealing package and the D-Wave . The numerical results very nicely compare with those obtained with standard classical algorithms, and also show interesting scalability features. Published by the American Physical Society 2024

Quantum-Annealing-Inspired Algorithms for Track Reconstruction at High-Energy Colliders

Charged particle reconstruction or track reconstruction is one of the most crucial components of pattern recognition in high-energy collider physics. It is known to entail enormous consumption of computing resources, especially when the particle multiplicity is high, which will be the conditions at future colliders, such as the High Luminosity Large Hadron Collider and Super Proton–Proton Collider. Track reconstruction can be formulated as a quadratic unconstrained binary optimization (QUBO) problem, for which various quantum algorithms have been investigated and evaluated with both a quantum simulator and hardware. Simulated bifurcation algorithms are a set of quantum-annealing-inspired algorithms, known to be serious competitors to other Ising machines. In this study, we show that simulated bifurcation algorithms can be employed to solve the particle tracking problem. The simulated bifurcation algorithms run on classical computers and are suitable for parallel processing and usage of graphical processing units, and they can handle significantly large amounts of data at high speed. These algorithms exhibit reconstruction efficiency and purity comparable to or sometimes improved over those of simulated annealing, but the running time can be reduced by as much as four orders of magnitude. These results suggest that QUBO models together with quantum-annealing-inspired algorithms are valuable for current and future particle tracking problems.

Modeling routing problems in QUBO with application to ride-hailing

Many emerging commercial services are based on the sharing or pooling of resources for common use with the aim of reducing costs. Businesses such as delivery-, mobility-, or transport-as-a-service have become standard in many parts of the world, fulfilling on-demand requests for customers in live settings. However, it is known that many of these problems are NP-hard, and therefore both modeling and solving them accurately is a challenge. Here we focus on one such routing problem, the Ride Pooling Problem (RPP), where multiple customers can request on-demand pickups and drop-offs from shared vehicles within a fleet. The combinatorial optimization task is to optimally pool customer requests using the limited set of vehicles, akin to a small-scale flexible bus route. In this work, we propose a quadratic unconstrained binary optimization (QUBO) program and introduce efficient formulation methods for the RPP to be solved using metaheuristics, and specifically emerging quantum optimization algorithms.

QUBO Models for the FIFO Stack-Up Problem and Experimental Evaluation on a Quantum Annealer

Quantum annealing has been applied to combinatorial optimization problems in recent years. In this paper we study the possibility to use quantum annealing for solving the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from a conveyor belt onto pallets. The problem is NP-hard and can be solved using linear programming approaches. We developed two QUBO (quadratic unconstrained binary optimization) objective functions based on a bin stack-up solution and a pallet stack-up solution for this problem suitable for a quantum annealer. The number of variables was minimized to increase the performance and their dependence on the number of bins and pallets was discussed. The performances of both methods were studied for various small problem sizes on a D-Wave quantum annealer. We found that only tiny instances could be solved and looked at the terms of the QUBO-formulations, which cause the quantum annealer to fail for larger problem sizes. Furthermore we compare the results to the performance of a classic computer using the same QUBO-formulations.

An integrated coupled oscillator network to solve optimization problems

Solving combinatorial optimization problems is essential in scientific, technological, and engineering applications, but can be very time and energy-consuming using classical algorithms executed on digital processors. Oscillator-based Ising machines offer a promising alternative by exploiting the analog coupling between electrical oscillators to solve such optimization problems more efficiently. Here we present the design and the capabilities of our scalable approach to solve Ising and quadratic unconstrained binary optimization problems. This approach includes routable oscillator connections to simplify the time-consuming embedding of the problem into the oscillator network. Our manufactured silicon chip, featuring 1440 oscillators implemented in a 28 nm technology, demonstrates the ability to solve optimization problems in 950 ns while consuming typically 319 μW per node. A frequency, phase, and delay calibration ensures robustness against manufacturing variations. The system is evaluated with multiple sets of benchmark problems to analyze the sensitivity for parameters such as the coupling strength or frequency.

Analyzing the effectiveness of quantum annealing with meta-learning

The field of Quantum Computing has gathered significant popularity in recent years and a large number of papers have studied its effectiveness in tackling many tasks. We focus in particular on Quantum Annealing (QA), a meta-heuristic solver for Quadratic Unconstrained Binary Optimization (QUBO) problems. It is known that the effectiveness of QA is dependent on the task itself, as is the case for classical solvers, but there is not yet a clear understanding of which are the characteristics of a problem that make it difficult to solve with QA. In this work, we propose a new methodology to study the effectiveness of QA based on meta-learning models. To do so, we first build a dataset composed of more than five thousand instances of ten different optimization problems. We define a set of more than a hundred features to describe their characteristics and solve them with both QA and three classical solvers. We publish this dataset online for future research. Then, we train multiple meta-models to predict whether QA would solve that instance effectively and use them to probe which features with the strongest impact on the effectiveness of QA. Our results indicate that it is possible to accurately predict the effectiveness of QA, validating our methodology. Furthermore, we observe that the distribution of the problem coefficients representing the bias and coupling terms is very informative in identifying the probability of finding good solutions, while the density of these coefficients alone is not enough. The methodology we propose allows to open new research directions to further our understanding of the effectiveness of QA, by probing specific dimensions or by developing new QUBO formulations that are better suited for the particular nature of QA. Furthermore, the proposed methodology is flexible and can be extended or used to study other quantum or classical solvers.