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. The authors present a method for creating generative models that produce test instances for quadratic unconstrained binary optimization (QUBO) problems, inspired by Hebb’s rule. Their approach identifies key solution patterns and complexity transitions, providing a platform to evaluate the performance of Ising hardware and solvers efficiently.
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. The authors present a method for creating generative models that produce test instances for quadratic unconstrained binary optimization (QUBO) problems, inspired by Hebb’s rule. Their approach identifies key solution patterns and complexity transitions, providing a platform to evaluate the performance of Ising hardware and solvers efficiently.
We introduce universal, easy-to-reproduce generative models for the QUBO instances to differentiate the performance of the hardware/solvers effectively. Our benchmark process extends the well-known Hebb's rule of associative memory with the asymmetric pattern weights. We provide a comprehensive overview of calculations conducted across various scales and using different classes of dynamical equations. Our aim is to analyze their results, including factors such as the probability of encountering the ground state, planted state, spurious state, or states falling outside the predetermined energy range. Moreover, the generated problems show additional properties, such as the easy-hard-easy complexity transition and complicated cluster structures of planted solutions. Our method establishes a prospective platform to potentially address other questions related to the fundamental principles behind device physics and algorithms for novel computing machines.
We are interested in benchmarking both quantum annealing and classical algorithms for minimizing quadratic unconstrained binary optimization (QUBO) problems. Such problems are NP-hard in general, implying that the exact minima of randomly generated instances are hard to find and thus typically unknown. While brute forcing smaller instances is possible, such instances are typically not interesting due to being too easy for both quantum and classical algorithms. In this contribution, we propose a novel method, called posiform planting, for generating random QUBO instances of arbitrary size with known optimal solutions, and use those instances to benchmark the sampling quality of four D-Wave quantum annealers utilizing different interconnection structures (Chimera, Pegasus, and Zephyr hardware graphs) and the simulated annealing algorithm. Posiform planting differs from many existing methods in two key ways. It ensures the uniqueness of the planted optimal solution, thus avoiding groundstate degeneracy, and it enables the generation of QUBOs that are tailored to a given hardware connectivity structure, provided that the connectivity is not too sparse. Posiform planted QUBOs are a type of 2-SAT boolean satisfiability combinatorial optimization problems. Our experiments demonstrate the capability of the D-Wave quantum annealers to sample the optimal planted solution of combinatorial optimization problems with up to 5, 627 qubits.
Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.
Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. As many NP problems can be reduced to QUBO problems, considerable research has gone into developing QUBO solvers running on various computing platforms such as quantum devices, ASICs, FPGAs, GPUs, and optical fibers. For evaluating the performance of QUBO solvers, QUBO problems reduced from combinatorial optimization problems such as MaxCut, Traveling Salesman Problem (TSP), and Quadratic Assignment Problem (QAP) are commonly used. This paper presents a new benchmark QUBO problem inspired by digital halftoning, which aims to find a binary image reproducing an input grayscale image. The characteristics of this QUBO problem reduced from digital halftoning are quite unique. Solutions of the QUBO problem can be converted into binary images, from which we can visually evaluate the solution quality. Good approximation solutions for the QUBO problem can be obtained by a simple heuristic algorithm very quickly. However, it is very hard to find an optimal solution. Moreover, quite interestingly, we can create a QUBO problem for digital halftoning with a known optimal solution for a benchmarking purpose. Thus, we can exactly compute the gap between the optimal solution and an approximate solution obtained by a QUBO solver. We have developed an OpenMP-based sparse QUBO solver that can find good approximate solutions for QUBO problems reduced from digital halftoning. We have evaluated the performance of QUBO solvers including the D-Wave Hybrid solver, Gurobi optimizer, Open.Jij, and our sparse QUBO solver. The experimental results show that our OpenMP-based sparse QUBO solver attains the best performance.
Quantum computing promises to have a tremendous advantage over its classical counterpart for solving computationally hard problems, yet remains in a relatively early stage for practical applications owing to the limited capabilities of today’s quantum computers. The approach to the special purposes of quantum computers by exploiting the special-purpose physics-inspired or quantum-inspired computers is emerging as a novel alternative to its quantum counterpart in tackling hard problems in high-performance computing. Inspired by physics, the Ising machine — a type of special-purpose computer that implements or emulates physics or quantum effects of the Ising model to speed up finding solutions to optimization problems — has recently become an active research area in the field of combinatorial optimization. This paper is to address the key enabling software and hardware technology underlying physics-inspired optimization using Ising machines in the unified quadratic unconstrained binary optimization (QUBO) framework for modeling and solving computationally hard combinatorial optimization problems, and with an aim to shed some light on the challenges and opportunities associated with the ever-growing landscape of this novel high-performance computing.
Abstract 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.