Quadratic Unconstrained Binary Optimization Research Articles

MO965: The Implementation of A Novel Algorithm for Optimal Predictor Selection, Exploiting Miqubo on D-Wave Quantum Annealer, for Predicting Kidney Graft Loss After Urinary Sepsis in Kidney Transplant Recipients

Abstract BACKGROUND AND AIMS Kidney transplant recipients (KTRs) hospitalized for urinary sepsis are at increased risk of graft loss (GL). Thus, there is a distinct need to develop models to accurately predict significant graft complications. MIQUBO (Mutual Information Quadratic Unconstrained Binary Optimization) is a concept, which in great approximation relies on optimization (maximization) of Mutual Information (the amount of information that is carried by a single variable) based on QUBO (Quadratic Unconstrained Binary Optimization) problem [1,2]. This method is found to be especially fit to solve complex statistical problems due to its ability to reduce models dimensionality, especially when deployed on a D-wave quantum computer [3]. In this study, we have investigated the feasibility of MIQUBO as a novel feature selection method to consciously extend machine learning prediction models, using a previously published dataset [4], to predict GL after urosepsis in KTRs. METHOD The analyzed dataset included 101 KTRs hospitalized for urosepsis, 100 KTRs hospitalized for UTI and 100 healthy KTRs without any history of UTI or sepsis. To predict GL at 12 months post-discharge after an episode of urosepsis, four features have been used to predict graft loss in a previous (unpublished) study by our group. These four features are further referred to as restricted variable set (RVS*). Subsequently, from the total of 150 variables included in the dataset, we have selected 12 additional features using MIQUBO deployed on a D-Wave quantum annealer. The extended variable set (EVS) was created based upon Conditional Mutual Information (CMI) of each variable. Then, 13 different machine learning (ML) models were build using each set of variables. Each developed model was explored for its performance according to the amount of included additional features (random versus MI-ranked). RESULTS The overall frequency of graft loss after 12 months of observation equaled 10.3% in the whole study cohort. However, the frequency of GL after urosepsis has reached 18% after 12 months from discharge, as compared with 3% in the healthy controls. The variables included in both sets of features (RVS and EVS) and their values were presented in Table 1. The best performance among developed models (using the EVS) has been achieved using Gaussian Process and QDA models, yielding area-under-the-receiver-operator-curve (AUROC) of approximately 0.91. From 13 developed pairs of models (for both sets of variables), 8 have proven to be statistically superior (AUROC comparison) in favor of MIQUBO-extended models. The mean difference of AUROCC between EVS- and RVS-models equaled 0.092 ± 0.11. Furthermore, a trend has been observed towards better performance of models (in regard of achieved AUC and recall) when ML-models were extended utilizing features based upon MIQUBO MI-ranking as compared with features randomly chosen from EVS. Representative plots were presented on Figure 1. The point ‘0’ on these plots refers to the parameters of models built on sole RVS. CONCLUSION Our preliminary experiment suggests that MIQUBO deployed on a quantum annealer is an efficient method for extending machine learning models with variables derived from extensive and imbalanced datasets, without losing on models’ reproducibility or accuracy.

Quantum Bridge Analytics II: QUBO-Plus, network optimization and combinatorial chaining for asset exchange.

Quantum Bridge Analytics relates to methods and systems for hybrid classical-quantum computing, and is devoted to developing tools for bridging classical and quantum computing to gain the benefits of their alliance in the present and enable enhanced practical application of quantum computing in the future. This is the second of a two-part tutorial that surveys key elements of Quantum Bridge Analytics and its applications. Part I focused on the Quadratic Unconstrained Binary Optimization (QUBO) model which is presently the most widely applied optimization model in the quantum computing area, and which unifies a rich variety of combinatorial optimization problems. Part II (the present paper) introduces the domain of QUBO-Plus models that enables a larger range of problems to be handled effectively. After illustrating the scope of these QUBO-Plus models with examples, we give special attention to an important instance of these models called the Asset Exchange Problem (AEP). Solutions to the AEP enable market players to identify exchanges of assets that benefit all participants. Such exchanges are generated by a combination of two optimization technologies for this class of QUBO-Plus models, one grounded in network optimization and one based on a new metaheuristic optimization approach called combinatorial chaining. This combination opens the door to expanding the links to quantum computing applications established by QUBO models through the Quantum Bridge Analytics perspective. We show how the modeling and solution capability for the AEP instance of QUBO-Plus models provides a framework for solving a broad range of problems arising in financial, industrial, scientific, and social settings.

GPU-accelerated scalable solver with bit permutated cyclic-min algorithm for quadratic unconstrained binary optimization

A wide range of combinatorial optimization problems can be reduced to the Ising model, and equivalently the quadratic unconstrained binary optimization (QUBO) problem. Thus, in recent years, researchers have proposed to solve QUBO on FPGAs, GPUs, and special-purpose processors. The adaptive bulk search (ABS) is a previously-proposed framework for solving QUBO in parallel on multiple GPUs. In the ABS, a CPU host performs a GA-based global search while GPUs asynchronously perform many local searches in parallel. An original ABS adopts a simple local search algorithm called a cyclic-min algorithm, which does not use pseudo random numbers. However, the lack of randomness may cause a potential drawback of restricted bit-flipping operations in a local search. To avoid this drawback, this paper proposes a cyclic-min algorithm with randomly-generated multiple bit permutations, which enables a more effective local search with random number generation in CPUs (not in GPUs). Furthermore, this paper introduces a scalable implementation of the ABS with MPI and OpenMP. Our experimental results on TSUBAME3.0 show that the solution quality improves and the throughput linearly increases as the number of GPUs increases; with 256 GPUs, it evaluates 20.1×1012 solutions per second.

Data structures for speeding up Tabu Search when solving sparse quadratic unconstrained binary optimization problems

Data structures for speeding up Tabu Search when solving sparse quadratic unconstrained binary optimization problems

Continuous black-box optimization with an Ising machine and random subspace coding

A black-box optimization algorithm such as Bayesian optimization finds the extremum of an unknown function by alternating the inference of the underlying function and optimization of an acquisition function. In a high-dimensional space, such algorithms perform poorly due to the difficulty of acquisition function optimization. Herein, we apply Ising machines to overcome the difficulty in the continuous black-box optimization. As an Ising machine specializes in optimization of binary problems, a continuous vector has to be encoded to binary, and the solution by Ising machines has to be translated back. Our method has the following three parts: (1) random subspace coding based on axis-parallel hyperrectangles from continuous vector to binary vector, (2) a quadratic unconstrained binary optimization (QUBO) defined by the acquisition function based on the nonnegative-weighted linear regression model, which is solved by Ising machines, and (3) a penalization scheme to ensure that the solution can be translated back. It is shown with benchmark tests that its performance using the D-Wave Advantage quantum annealer and simulated annealing is competitive with a state-of-the-art method based on the Gaussian process in high-dimensional problems. Our method may open up the possibility of Ising machines and other QUBO solvers, including a quantum approximate optimization algorithm using gated-quantum computers, and may expand its range of application to continuous-valued problems.

Goal seeking Quadratic Unconstrained Binary Optimization

The Quadratic Unconstrained Binary Optimization (QUBO) modeling and solution framework is a requirement for quantum and digital annealers. However optimality for QUBO problems of any practical size is extremely difficult to achieve. In order to incorporate the problem-specific insights, a diverse set of solutions meeting an acceptable target metric or goal is the preference in high level decision making. In this paper, we present two alternatives for goal-seeking QUBO for minimizing the deviation from a given target as well as a range of values around a target. Experimental results illustrate the efficacy of the proposed approach over Constraint Programming for quickly finding a satisficing set of solutions.

Combinatorial optimization with physics-inspired graph neural networks

Combinatorial optimization problems are pervasive across science and industry. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that incorporates insights from statistical physics is still outstanding. Here we demonstrate how graph neural networks can be used to solve combinatorial optimization problems. Our approach is broadly applicable to canonical NP-hard problems in the form of quadratic unconstrained binary optimization problems, such as maximum cut, minimum vertex cover, maximum independent set, as well as Ising spin glasses and higher-order generalizations thereof in the form of polynomial unconstrained binary optimization problems. We apply a relaxation strategy to the problem Hamiltonian to generate a differentiable loss function with which we train the graph neural network and apply a simple projection to integer variables once the unsupervised training process has completed. We showcase our approach with numerical results for the canonical maximum cut and maximum independent set problems. We find that the graph neural network optimizer performs on par or outperforms existing solvers, with the ability to scale beyond the state of the art to problems with millions of variables.

Compressed sensing based on QUBO formulation

Ising machines efficiently solve the combinatorial optimization problems described by the Ising model or the quadratic unconstrained binary optimization (QUBO) formulation. A hybrid method based on the QUBO formulation for compressed sensing is proposed. The proposed method comprises alternative steps of discrete and continuous optimization. In the discrete optimization step, the objective function is described by the QUBO formulation. Successful examples obtained via the proposed method are demonstrated. The performance of the proposed method depends highly on the initial conditions.

Leveraging special-purpose hardware for local search heuristics

As we approach the physical limits predicted by Moore’s law, a variety of specialized hardware is emerging to tackle specialized tasks in different domains. Within combinatorial optimization, adiabatic quantum computers, complementary metal-oxide semiconductor annealers, and optical parametric oscillators are a few emerging specialized hardware technologies to solve optimization problems. The Ising optimization model unifies all of these emerging special-purpose hardware for optimization in terms of mathematical framework. In other words, they are all designed to solve optimization problems expressed in the Ising model or equivalently as a quadratic unconstrained binary optimization model. Due to various constraints specific to each type of hardware, they usually suffer from a major challenge: the number of variables that the hardware can manage to solve is very limited. The local search meta-heuristic is one of the approaches to tackle large-scale problems. However, a general optimization step within local search is not traditionally formulated in the Ising form. In this work, we introduce a new modeling framework for modeling local search heuristics for special-purpose hardware. In particular, we propose models that take the limitations of the Ising model and current hardware into account. As such, we demonstrate the advantage of our approach compared to previous methods by carrying out experiments to show that our local search models produce higher-quality solutions.

Characterization of QUBO reformulations for the maximum k-colorable subgraph problem

Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem’s constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out affects the value of the penalty parameters, as well as the number of additional binary variables that are needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect the ability of quantum computers to efficiently solve these constrained COPT problems. This efficiency is the key toward the goal of using quantum computers to solve constrained COPT problems more efficiently than with classical computers. Along these lines, we consider an important constrained COPT problem, namely the maximum k-colorable subgraph (MkCS) problem, in which the aim is to find an induced k-colorable subgraph with maximum cardinality in a given graph. This problem arises in channel assignment in spectrum sharing networks, VLSI design, human genetic research, and cybersecurity. We derive two QUBO reformulations for the MkCS problem and fully characterize the range of the penalty parameters that can be used in the QUBO reformulations. Further, one of the QUBO reformulations of the MkCS problem is obtained without the need to introduce additional binary variables. To illustrate the benefits of obtaining and characterizing these QUBO reformulations, we benchmark different QUBO reformulations of the MkCS problem by performing numerical tests on D-Wave’s quantum annealing devices. These tests also illustrate the numerical power gained by using the latest D-Wave’s quantum annealing device.

Distance-based clustering using QUBO formulations

In computer science, clustering is a technique for grouping data. Ising machines can solve distance-based clustering problems described by quadratic unconstrained binary optimization (QUBO) formulations. A typical simple method using an Ising machine makes each cluster size equal and is not suitable for clustering unevenly distributed data. We propose a new clustering method that provides better performance than the simple method, especially for unevenly distributed data. The proposed method is a hybrid algorithm including an iterative process that comprises solving a discrete optimization problem with an Ising machine and calculating parameters with a general-purpose computer. To minimize the communication overhead between the Ising machine and the general-purpose computer, we employed a low-latency Ising machine implementing the simulated bifurcation algorithm with a field-programmable gate array attached to a local server. The proposed method results in clustering 200 unevenly distributed data points with a clustering score 18% higher than that of the simple method. The discrete optimization with 2000 variables is performed 100 times per iteration, and the overhead time is reduced to approximately 20% of the total execution time. These results suggest that hybrid algorithms using Ising machines can efficiently solve practical optimization problems.

3-regular three-XORSAT planted solutions benchmark of classical and quantum heuristic optimizers

With current semiconductor technology reaching its physical limits, special-purpose hardware has emerged as an option to tackle specific computing-intensive challenges. Optimization in the form of solving quadratic unconstrained binary optimization problems, or equivalently Ising spin glasses, has been the focus of several new dedicated hardware platforms. These platforms come in many different flavors, from highly-efficient hardware implementations on digital-logic of established algorithms to proposals of analog hardware implementing new algorithms. In this work, we use a mapping of a specific class of linear equations whose solutions can be found efficiently, to a hard constraint satisfaction problem (three-regular three-XORSAT, or an Ising spin glass) with a ‘golf-course’ shaped energy landscape, to benchmark several of these different approaches. We perform a scaling and prefactor analysis of the performance of Fujitsu’s digital annealer unit (DAU), the D-Wave advantage quantum annealer, a virtual MemComputing machine, Toshiba’s simulated bifurcation machine (SBM), the SATonGPU algorithm from Bernaschi et al, and our implementation of parallel tempering. We identify the SATonGPU and DAU as currently having the smallest scaling exponent for this benchmark, with SATonGPU having a small scaling advantage and in addition having by far the smallest prefactor thanks to its use of massive parallelism. Our work provides an objective assessment and a snapshot of the promise and limitations of dedicated optimization hardware relative to a particular class of optimization problems.

Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization

Recently, inspired by quantum annealing, many solvers specialized for unconstrained binary quadratic programming problems have been developed. For further improvement and application of these solvers, it is important to clarify the differences in their performance for various types of problems. In this study, the performance of four quadratic unconstrained binary optimization problem solvers, namely D-Wave Hybrid Solver Service (HSS), Toshiba Simulated Bifurcation Machine (SBM), Fujitsu Digital Annealer (DA), and simulated annealing on a personal computer, was benchmarked. The problems used for benchmarking were instances of real problems in MQLib, instances of the SAT-UNSAT phase transition point of random not-all-equal 3-SAT (NAE 3-SAT), and the Ising spin glass Sherrington-Kirkpatrick (SK) model. Concerning MQLib instances, the HSS performance ranked first; for NAE 3-SAT, DA performance ranked first; and regarding the SK model, SBM performance ranked first. These results may help understand the strengths and weaknesses of these solvers.

GPS: A New TSP Formulation for Its Generalizations Type QUBO

We propose a new Quadratic Unconstrained Binary Optimization (QUBO) formulation of the Travelling Salesman Problem (TSP), with which we overcame the best formulation of the Vehicle Routing Problem (VRP) in terms of the minimum number of necessary variables. After, we will present a detailed study of the constraints subject to the new TSP model and benchmark it with MTZ and native formulations. Finally, we will test whether the correctness of the formulation by entering it into a QUBO problem solver. The solver chosen is a D-Wave_2000Q6 quantum computer simulator due to the connection between Quantum Annealing and QUBO formulations.

Unconstrained binary models of the travelling salesman problem variants for quantum optimization

Quantum computing is offering a novel perspective for solving combinatorial optimization problems. To fully explore the possibilities offered by quantum computers, the problems need to be formulated as unconstrained binary models, taking into account limitation and advantages of quantum devices. In this work, we provide a detailed analysis of the travelling salesman problem with time windows (TSPTW) in the context of solving it on a quantum computer. We introduce quadratic unconstrained binary optimization and higher-order binary optimization formulations of this problem. We demonstrate the advantages of edge-based and node-based formulations of the TSPTW problem. Additionally, we investigate the experimental realization of the presented methods on a quantum annealing device. The provided results pave the path for utilizing quantum computer for a variety of real-world tasks which can be cast in the form of travelling salesman problem with time windows.

Converting of Boolean Expression to Linear Equations, Inequalities and QUBO Penalties for Cryptanalysis

There exists a wide range of constraint programming (CP) problems defined on Boolean functions depending on binary variables. One of the approaches to solving CP problems is using specific appropriate solvers, e.g., SAT solvers. An alternative is using the generic solvers for mixed-integer linear programming problems (MILP), but they require transforming expressions with Boolean functions to linear equations or inequalities. Here, we present two methods of such a transformation which applies to any Boolean function defined by explicit rules giving values of the Boolean function for all combinations of its Boolean variables. The first method represents the Boolean function as a linear equation in the original binary variables and, possibly, binary ancillaries, which become additional variables of the MILP problem being composed. The second method represents the Boolean function as a set of linear inequalities in the original binary variables and one additional continuous variable (representing the value of the function). The choice between the first or second method is a trade-off between the number of binary variables and number of linear constraints in the emerging MP problem. The advantage of the proposed approach is that both methods reduce important cryptanalysis problems, such as the preimaging of hash functions or breaking symmetric ciphers as the MILP problems, which are solved by the generic MILP solvers. Furthermore, the first method enables to reduce the binary linear equations to quadratic unconstrained binary optimization (QUBO), by the quantum annealer, e.g., D-Wave.

Exploiting Hardware and Software Advances for Quadratic Models of Wind Farm Layout Optimization

A key aspect of the design of a wind farm is the wind farm layout optimization (WFLO) problem: given a wind farm site and information about the wind patterns, the problem is to decide the location of individual wind turbines to maximize energy production subject to proximity restrictions and wake-based interference between turbines. Given the pairwise wake interactions, it is natural to model the energy objective as a quadratic function as, indeed, has been done in some existing optimization approaches. However, state-of-the-art solutions often trade-off between speed in producing designs and quality in terms of finding and proving optimal solutions. In this work, we aim to find a balanced approach to obtain WFLO solutions quickly for interactive design and solve the problem to optimality when quality is more important. To this end, we exploit recent advances in optimization hardware and software that target quadratic constraints: commercial mixed integer linear solvers have been extended to address some quadratic problems and nascent specialized hardware, including quantum computing systems, have focused on solving quadratic unconstrained binary optimization (QUBO) problems. We introduce two novel quadratic programming models for WFLO: a quadratic constrained optimization problem (QCOP) with binary decision variables and a QUBO. A thorough numerical evaluation using a commercial solver and specialized QUBO hardware show that our quadratic framework achieves fast, high-quality solutions that improve the state of the art and strike a balance between speed and quality. In particular, the QUBO model delivers high quality solutions in a few seconds while the QCOP model can be used to find better solutions and provide quality guarantees over a longer run-time.

Quantum Annealing Methods and Experimental Evaluation to the Phase-Unwrapping Problem in Synthetic Aperture Radar Imaging

The focus of this work is to explore the use of quantum annealing solvers for the problem of phase unwrapping of synthetic aperture radar (SAR) images. Although solutions to this problem exist based on network programming, these techniques do not scale well to larger sized images. Our approach involves formulating the problem as a quadratic unconstrained binary optimization (QUBO) problem, which can be solved on a quantum annealer. Given that present embodiments of quantum annealers remain limited in the number of qubits they possess, we decompose the problem into a set of subproblems that can be solved individually. These individual solutions are close to optimal up to an integer constant, with one constant per subimage. In a second phase, these integer constants are determined as a solution to yet another QUBO problem. This basic idea is extended to several passes, where each pass results in an image which is subsequently decomposed to yet another set of subproblems until the resulting image can be accommodated by the annealer at hand. Additionally, we explore improvements to the method by decomposing the original image into overlapping subimages and ignoring the results on the overlapped (marginal) pixels. We test our approach with a variety of software-based QUBO solvers and on a variety of images, both synthetic and real. Additionally, we experiment using D-wave systems’ quantum annealer, the D-wave 2000Q_6 and developed an embedding method which, for our problem, yielded improved results. Our method resulted in high quality solutions, comparable to state-of-the-art phase-unwrapping solvers.

New Results on Closed-form Formulas for Evaluating r-flip Moves in Quadratic Unconstrained Binary Optimization

New Results on Closed-form Formulas for Evaluating r-flip Moves in Quadratic Unconstrained Binary Optimization

Deep Space Network Scheduling Using Quantum Annealing

NASA's Deep Space Network (DSN) is responsible for communication and navigation for several NASA and international missions. The DSN comprises three complexes located in Goldstone (California, USA), Cambera (Australia) and Madrid (Spain). This distribution in longitude guarantees a full sky coverage. Each complex has one 70-meter and several 34-meter antennas. The network routinely serves a few dozens missions. The scheduling of the DSN is complex and involves humans interventions as well as automated solutions. In order to increase the level of automation, different computing paradigms have been explored. In this paper, we report on the quadratic unconstrained binary optimization (QUBO) formulation of the DSN scheduling, as well as a custom classical solver designed around some of the unique features of this scheduling problem. Thanks to a hybrid framework that extends the size of the problems that can be solved with a quantum annealer, we are able to generate a schedule from the QUBO formulation of the problem for one week's worth of user antenna requests, which represents the time period scheduled during operation. In other words, this work describes a real- world application of quantum annealing using real-world, operational data. We compare the resulting schedules' quality to solutions obtained using a mixed-integer linear programming formulation on a commercial solver. Our custom solver, based on a quantum-inspired optimization technique called Substochatic Monte-Carlo, while much faster in generating schedules, could only treat a subset of requests and hence we report its results independently.