Black-box optimization using factorization and Ising machines

We are in the noisy intermediate-scale quantum (NISQ) devices' era, in which quantum hardware has become available for application in real-world problems. However, demonstrations of the usefulness of such NISQ devices are still rare. In this work, we consider a practical railway dispatching problem: delay and conflict management on single-track railway lines. We examine the train dispatching consequences of the arrival of an already delayed train to a given network segment. This problem is computationally hard and needs to be solved almost in real time. We introduce a quadratic unconstrained binary optimization (QUBO) model of this problem, which is compatible with the emerging quantum annealing technology. The model's instances can be executed on present-day quantum annealers. As a proof-of-concept, we solve selected real-life problems from the Polish railway network using D-Wave quantum annealers. As a reference, we also provide solutions calculated with classical methods, including the conventional solution of a linear integer version of the model as well as the solution of the QUBO model using a tensor network-based algorithm. Our preliminary results illustrate the degree of difficulty of real-life railway instances for the current quantum annealing technology. Moreover, our analysis shows that the new generation of quantum annealers (the advantage system) does not perform well on those instances, either.

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.

Low-density parity-check (LDPC) codes have previously been considered as combinatorial optimization problems (COPs) in respect to its decoding. However, after defining it as such, none have gone so far as to convert the LDPC code into a quadratic unconstrained binary optimization (QUBO) problem. Thus, a new method is created: one that converts the LDPC code to a QUBO problem, inputs the QUBO problem into Ising machines (computers based on the Ising model that are designed to solve the QUBO problem), obtains the QUBO solution and converts it to a LDPC solution. By utilizing an actual Ising machine, LDPC solutions with code length of 256-bits have been obtained with an accuracy of 93.9% by average annealing time 214.0ms. The benefit of this newfound methodology goes beyond its theoretical imprint of obtaining LDPC solutions more accurately. It has only been a few years since the Ising machine has been developed. Therefore, in formulating this method, one expands the currently scope of studies involving Ising machines, helping current and future researchers unlock its full range of capabilities and possibilities.

In edge computing, suppressing data size is a challenge for machine learning models that perform complex tasks such as autonomous driving, in which computational resources (speed, memory size and power) are limited. Efficient lossy compression of matrix data has been introduced by decomposing it into the product of an integer and real matrices. However, its optimisation is difficult as it requires simultaneous optimisation of an integer and real variables. In this paper, we improve this optimisation by utilising recently developed black-box optimisation (BBO) algorithms with an Ising solver for binary variables. In addition, the algorithm can be used to solve mixed-integer programming problems that are linear and non-linear in terms of real and integer variables, respectively. The differences between the choice of Ising solvers (simulated annealing, quantum annealing and simulated quenching) and the strategies of the BBO algorithms (BOCS, FMQA and their variations) are discussed for further development of the BBO techniques.

Although many efficient heuristics have been developed to solve binary optimization problems, these typically produce correlated solutions for degenerate problems. Most notably, transverse-field quantum annealing-the heuristic employed in current commercially available quantum annealing machines-has been shown to often be exponentially biased when sampling the solution space. Here we present an approach to sample ground-state (or low-energy) configurations for binary optimization problems. The method samples degenerate states with almost equal probability and is based on a combination of parallel tempering Monte Carlo with isoenergetic cluster moves. We illustrate the approach using two-dimensional Ising spin glasses, as well as spin glasses on the D-Wave Systems quantum annealer chimera topology. In addition, a simple heuristic to approximate the number of solutions of a degenerate problem is introduced.

Factorization Machines (FMs) are a series of effective solutions for sparse data prediction by considering the interactions among users, items, and auxiliary information. However, the feature representations in most state-of-the-art FMs are fixed, which reduces the prediction performance as the same feature may have unequal predictabilities under different input instances. In this paper, we propose a novel Feature-adjusted Factorization Machine (FaFM) model by adaptively adjusting the feature vector representations from both vector-level and bit-level. Specifically, we adopt a fully connected layer to adaptively learn the weight of vector-level feature adjustment. And a user-item specific gate is designed to refine the vector in bit-level and to filter noises caused by over-adaptation of the input instance. Extensive experiments on two real-world datasets demonstrate the effectiveness of FaFM. Empirical results indicate that FaFM significantly outperforms the traditional FM with a 10.89% relative improvement in terms of Root Mean Square Error (RMSE) and consistently exceeds four state-of-the-art deep learning based models.

Quantum computations are very important branch of modern cryptology. According to the number of working physical qubits available in general-purpose quantum computers and in quantum annealers, there is no coincidence, that nowadays quantum annealers allow to solve larger problems. In this paper we focus on solving discrete logarithm problem (DLP) over binary fields using quantum annealing. It is worth to note, that however solving DLP over prime fields using quantum annealing has been considered before, no author, until now, has considered DLP over binary fields using quantum annealing. Therefore, in this paper, we aim to bridge this gap. We present a polynomial transformation of the discrete logarithm problem over binary fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem, using approximately 3n2 logical variables for the binary field F2n. In our estimations, we assume the existence of an optimal normal base in the given fields. Such a QUBO instance can then be solved using quantum annealing.

We propose a new kernel that quantifies success for the task of computing a core-periphery partition for an undirected network. Finding the associated optimal partitioning may be expressed in the form of a quadratic unconstrained binary optimization (QUBO) problem, to which a state-of-the-art quantum annealer may be applied. We therefore make use of the new objective function to (a) judge the performance of a quantum annealer, and (b) compare this approach with existing heuristic core-periphery partitioning methods. The quantum annealing is performed on a commercially available D-Wave machine. The QUBO problem involves a full matrix even when the underlying network is sparse. Hence, we develop and test a sparsified version of the original QUBO which increases the available problem dimension for the quantum annealer. Results are provided on both synthetic and real data sets, and we conclude that the QUBO/quantum annealing approach offers benefits in terms of optimizing this new quantity of interest.

With the slowdown of improvement in conventional von Neumann systems, increasing attention is paid to novel paradigms such as Ising machines. They have very different approach to solving combinatorial optimization problems. Ising machines have shown great potential in solving binary optimization problems like MaxCut. In this paper, we present an analysis of these systems in boolean satisfiability (SAT) problems. We demonstrate that, in the case of 3-SAT, a basic architecture fails to produce meaningful acceleration, largely due to the relentless progress made in conventional SAT solvers. Nevertheless, careful analysis attributes part of the failure to the lack of two important components: cubic interactions and efficient randomization heuristics. To overcome these limitations, we add proper architectural support for cubic interaction on a state-of-the-art Ising machine. More importantly, we propose a novel semantic-aware annealing schedule that makes the search-space navigation much more efficient than existing annealing heuristics. Using numerical simulations, we show that such an “Augmented” Ising Machine for SAT is projected to outperform state-of-the-art software-based, GPU-based and conventional hardware SAT solvers by orders of magnitude.

Short-term GPS data based taxi pick-up area recommendation can improve the efficiency and reduce the overheads. But how to alleviate sparsity and further enhance accuracy is still challenging. Addressing at these issues, we propose to fuse spatio-temporal contexts into deep factorization machine (STC_DeepFM) offline for pick-up area recommendation, and within the area to recommend pick-up points online using factorization machine (FM). Firstly, we divide the urban area into several grids with equal size. Spatio-temporal contexts are destilled from pick-up points or points-of-interest (POIs) belonged to the preceding grids. Secondly, the contexts are integrated into deep factorization machine (DeepFM) to mine high-order interaction relationships from grids. And a novel algorithm named STC_DeepFM is presented for offline pick-up area recommendation. Thirdly, we devise the architecture of offline-to-online (O2O) recommendation respectively based on DeepFM and FM model in order to tradeoff the accuracy and efficiency. Some experiments are designed on the DiDi dataset to evaluate step by step the performance of spatio-temporal contexts, different recommendation models, and the O2O architecture. The results show that the proposed STC_DeepFM algorithm exceeds several state-of-the-art methods, and the O2O architecture achieves excellent real-time performance.

Recent hardware advances in quantum and quantum-inspired annealers promise substantial speedup for solving NP-hard combinatorial optimization problems compared to general-purpose computers. These special-purpose hardware are built for solving hard instances of Quadratic Unconstrained Binary Optimization (QUBO) problems. In terms of number of variables and precision of these hardware are usually resource-constrained and they work either in Ising space {-1,1} or in Boolean space {0,1}. Many naturally occurring problem instances are higher-order in nature. The known method to reduce the degree of a higher-order optimization problem uses Rosenberg's polynomial. The method works in Boolean space by reducing the degree of one term by introducing one extra variable. In this work, we prove that in Ising space the degree reduction of one term requires the introduction of two variables. Our proposed method of degree reduction works directly in Ising space, as opposed to converting an Ising polynomial to Boolean space and applying previously known Rosenberg's polynomial. For sparse higher-order Ising problems, this results in a more compact representation of the resultant QUBO problem, which is crucial for utilizing resource-constrained QUBO solvers.

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.

Earthquake rupture forecasting is a critical component of seismic hazard analysis and requires identifying physically plausible rupture paths across complex fault networks. At its core, this task could be formulated as a large-scale combinatorial optimization problem, which involves selecting an optimal subset of fault segments from a set of candidates. Such problems pose significant challenges for traditional algorithms, as the number of admissible multi-fault ruptures grows combinatorially. Current operational workflows rely on locally applied plausibility filters. While computationally efficient, such greedy local heuristics risk excluding globally competitive rupture scenarios, particularly in regions with dense fault connectivity and competing rupture pathways. This work investigates whether quantum annealing hardware can serve as a sampling accelerator for exploring the ensemble of physically plausible ruptures beyond what is currently accessible to classical approaches. Designing a problem formulation that remains scientifically meaningful while respecting the constraints of current quantum hardware (size, noise, etc.) is nontrivial, and naïve encodings often collapse under these limitations. We encode rupture plausibility modeling as low energy solutions to a quadratic unconstrained binary optimization (QUBO) problem defined on a fault-network middle graph. Binary variables represent activated fault segments, while local interaction terms encode Coulomb stress transfer (as a proxy for pairwise rupture compatibility), continuity preferences, and branching behavior. Here, we adopt a classical-quantum hybrid workflow: the quantum annealer is used to sample globally competitive rupture candidates, while classical post-processing implements physical constraints to filter out non-physical ruptures. The workflow is demonstrated on a subset of the fault network used in the 2023 USGS National Seismic Hazard Model including over 200 fault segments, using both simulated thermal and quantum annealing on D-Wave hardware. Results show that our hardware-aware formulation and conditioning enable robust sampling on comparatively large fault-network instances. Views on quantum computing are polarized: some overstate its power, while others dismiss it as impractical. Our results help bridge this by establishing a concrete, testable pathway for integrating quantum annealing into rupture-modeling workflows on existing purpose-built quantum devices.—---This work is supported by the ICSC National Research Centre for High Performance Computing, Big Data and Quantum Computing (CN00000013, CUP D53C22001300005) within the European Union-NextGenerationEU program (National Recovery and Resilience Plan (PNRR) - Mission 4 Component 2 Investment 1.4.)

In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), called ProCrastinated Tree Search (PCTS), that flexibly accommodates a delay and noise-tolerant bandit algorithm. We provide a generic proof technique to quantify regret of PCTS under delayed, noisy, and multi-fidelity feedbacks. Specifically, we derive regret bounds of PCTS enabled with delayed-UCB1 (DUCB1) and delayed-UCB-V (DUCBV) algorithms. Given a horizon T, PCTS retains the regret bound of non-delayed HOO for expected delay of O(log T), and worsens by T^((1-α)/(d+2)) for expected delays of O(T^(1-α)) for α ∈ (0,1]. We experimentally validate on multiple synthetic functions and hyperparameter tuning problems that PCTS outperforms the state-of-the-art black-box optimization methods for feedbacks with different noise levels, delays, and fidelity.

Abstract: This chapter explores the transformative potential of quantum algorithms in solving complex optimization problems across industries such as logistics, finance, and material science. Quantum algorithms, including Quantum Annealing, Quantum Approximate Optimization Algorithm (QAOA), and Variational Quantum Eigensolver (VQE), offer unprecedented computational capabilities for addressing challenges that are impractical for classical algorithms. In logistics, quantum algorithms optimize supply chains, vehicle routing, and warehouse operations, significantly reducing costs and improving efficiency. In finance, quantum optimization enables better portfolio management, risk assessment, and high-frequency trading strategies, providing faster, more accurate solutions. In material science, quantum algorithms accelerate the discovery and design of new materials, such as superconductors and catalysts, by simulating molecular interactions more efficiently. The chapter discusses real-world implementations, current technical challenges, and the future potential of quantum algorithms to revolutionize industries through faster and more scalable optimization solutions. Keywords: Quantum algorithms, Quantum Annealing, QAOA, VQE, optimization, logistics, finance, material science, portfolio optimization, risk management, supply chain, molecular simulation, quantum computing, quantum optimization, quantum chemistry, high-frequency trading.