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arXiv:2403.01320 · Physical review. E

Energy landscapes of combinatorial optimization in Ising machines

Dmitrii Dobrynin, Adrien Renaudineau, Mohammad Hizzani, Dmitri B. Strukov, Masoud Mohseni, John Paul Strachan

Published Oct 18, 2024• 19 citations
arXivHTMLPDF

Abstract

Physics-based Ising machines (IM) have been developed as dedicated processors for solving hard combinatorial optimization problems with higher speed and better energy efficiency. Generally, such systems employ local search heuristics to traverse energy landscapes in searching for optimal solutions. Here, we quantify and address some of the major challenges met by IMs by extending energy-landscape geometry visualization tools known as disconnectivity graphs. Using efficient sampling methods, we visually capture landscapes of problems having diverse structure and hardness manifesting as energetic and entropic barriers for IMs. We investigate energy barriers, local minima, and configuration space clustering effects caused by locality reduction methods when embedding combinatorial problems to the Ising hardware. To this end, we sample disconnectivity graphs of PUBO energy landscapes and their different QUBO mappings accounting for both local minima and saddle regions. We demonstrate that QUBO energy-landscape properties lead to the subpar performance for quadratic IMs and suggest directions for their improvement.

DOI
10.1103/physreve.110.045308

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Physics-based Ising machines (IM) have been developed as dedicated processors for solving hard combinatorial optimization problems with higher speed and better energy efficiency. Generally, such systems employ local search heuristics to traverse energy landscapes in searching for optimal solutions. Here, we quantify and address some of the major challenges met by IMs by extending energy-landscape geometry visualization tools known as disconnectivity graphs. Using efficient sampling methods, we visually capture landscapes of problems having diverse structure and hardness manifesting as energetic and entropic barriers for IMs. We investigate energy barriers, local minima, and configuration space clustering effects caused by locality reduction methods when embedding combinatorial problems to the Ising hardware. To this end, we sample disconnectivity graphs of PUBO energy landscapes and their different QUBO mappings accounting for both local minima and saddle regions. We demonstrate that QUBO energy-landscape properties lead to the subpar performance for quadratic IMs and suggest directions for their improvement.
Ising machines (IM) are physics-inspired alternatives to von Neumann architectures for solving hard optimization tasks. By mapping binary variables to coupled Ising spins, IMs can naturally solve unconstrained combinatorial optimization problems such as finding maximum cuts in graphs. However, despite their importance in practical applications, constrained problems remain challenging to solve for IMs that require large quadratic energy penalties to ensure the correspondence between energy ground states and constrained optimal solutions. To relax this requirement, we propose a self-adaptive IM that iteratively shapes its energy landscape using a Lagrange relaxation of constraints and avoids prior tuning of penalties. Using a probabilistic-bit (p-bit) IM emulated in software, we benchmark our algorithm with multidimensional knapsack problems (MKP) and quadratic knapsack problems (QKP), the latter being an Ising problem with linear constraints. For QKP with 300 variables, the proposed algorithm finds better solutions than state-of-the-art IMs such as Fujitsu's Digital Annealer and requires 7,500x fewer samples. Our results show that adapting the energy landscape during the search can speed up IMs for constrained optimization.
The key limitation of physics-inspired hardware accelerators such as Ising machines (IM) or Hopfield neural networks (HNN) is their lack of support of higher order couplings of spins/neurons. As a result, many classes of combinatorial optimization problems featuring polynomial interactions (PUBO) require quadratic hardware embeddings, QUBO mappings. Such embeddings were shown to risk severe performance reduction due to mismatched configuration space features (energy landscape) compared to the native space formulation. Here, we propose a gradient matching (GM) algorithm which leverages existing IM or in-memory HNN architectures and approximately recovers the native energy landscape. We show an exponential scaling advantage of the GM algorithm time-to-solution. Finally, we propose a hardware design to physically realize our GM in an HNN-based solver.
Ising Machines are emerging hardware architectures that efficiently solve NP-Hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ~10000x performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4x over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.
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.
Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary noisy intermediate-scale quantum (NISQ) era. The quantum approximate optimization algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of L quantum circuit layers within a corresponding Trotterized ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here we employ basin-hopping global optimization methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyze QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualizations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organization, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimization. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima. Published by the American Physical Society 2024
Abstract Ising Machines are emerging hardware architectures that efficiently solve NP-hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ∼\sim∼ ~ 10000 ×\times× × performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5 −-− − 4 ×\times× × over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.