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arXiv:2107.04304 · Applied Sciences

Petri Net Modeling for Ising Model Formulation in Quantum Annealing

Morikazu Nakamura, Kohei Kaneshima, Takeo Yoshida

Published Aug 18, 2021• 7 citations
arXivHTMLPDF

Abstract

Quantum annealing is an emerging new platform for combinatorial optimization, requiring an Ising model formulation for optimization problems. The formulation can be an essential obstacle to the permeation of this innovation into broad areas of everyday life. Our research is aimed at the proposal of a Petri net modeling approach for an Ising model formulation. Although the proposed method requires users to model their optimization problems with Petri nets, this process can be carried out in a relatively straightforward manner if we know the target problem and the simple Petri net modeling rules. With our method, the constraints and objective functions in the target optimization problems are represented as fundamental characteristics of Petri net models, extracted systematically from Petri net models, and then converted into binary quadratic nets, equivalent to Ising models. The proposed method can drastically reduce the difficulty of the Ising model formulation.

DOI
10.3390/app11167574

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Quantum annealing is a model for quantum computing that is aimed at solving hard optimization problems by representing them as quadratic unconstrained binary optimization (QUBO) problems. Although many NP-hard problems can easily be formulated as binary-variable problems with a quadratic objective function, such formulations typically also include constraints, which are not allowed in a QUBO. Hence, such constraints are usually incorporated in the objective function as additive penalty terms. While there is substantial previous work on implementing linear equality constraints, the case of inequality constraints has not much been studied. In this paper, we propose a new approach for formulating and embedding inequality constraints as penalties and describe early implementation results.
Recently, Ising machines have garnered attention as accelerators for resolving combinatorial optimization problems by converting them into Quadratic Unconstrained Binary Optimization (QUBO) problems. Ising machines have been rapidly developed with various implementations. For instance, there are different Ising machines like quantum annealing machines that use analog quantum circuits, simulated annealing machines that use digital processors like CPU, GPU, and VE, and digital annealing machines that use digital circuits such as CMOS and FPGA. However, the characteristics of Ising machines have not been fully comprehended, because a standardized evaluation method and a commonly used benchmark program for Ising machines have not been established yet. This research focuses on combinatorial clustering as a benchmark program for the evaluation of Ising machines, aiming to investigate the characteristics of Ising machines.
This paper explores the application of Quantum Annealing (QA) to the field of combinatorial optimization. The core idea is to leverage the principles of quantum mechanics, specifically quantum tunneling, to efficiently solve complex optimization problems. We present a framework for utilizing QA to tackle problems represented as quadratic unconstrained binary optimization (QUBO) models. The methodology focuses on mapping the problem's constraints and objective function into a form suitable for QA hardware. We discuss the challenges involved in this mapping process, including the limitations of current QA devices and the need for careful problem formulation. Furthermore, we analyze the potential advantages of QA over classical optimization techniques, particularly concerning the exploration of a wider solution space. The research highlights the promise of QACO as a viable approach for tackling computationally demanding combinatorial optimization tasks, offering a novel method to enhance optimization efficiency.
Quantum annealing computers are designed to produce high-quality solutions to optimization problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. While most of the well known NP-hard problems can easily be represented as quadratic binary problems, such formulations usually contain constraints that have to be added as penalties to the objective function in order to obtain QUBOs. In this paper, we propose a method based on finite automaton representation of the constraints for generating penalty implementations for them, which uses fewer qubits than the alternatives and is general enough to be applied to a whole class of constraints.
This paper investigates the application of quantum annealing to address the challenges of combinatorial optimization within large-scale network routing. Traditional classical algorithms frequently encounter significant computational bottlenecks when attempting to determine optimal routes across complex networks, often failing to scale effectively with increasing network size and complexity. Quantum annealing offers a potentially transformative approach by leveraging quantum mechanical effects to explore solution spaces more efficiently. This work outlines a methodology for mapping network routing problems – characterized by a vast number of possible routes – onto the quantum annealing hardware. We present a model for representing the problem, formulating it as an Ising model, and then detail the annealing process. The core claim is that quantum annealing can provide a viable solution for optimizing network routes, particularly in scenarios where classical methods are computationally intractable. We explore the underlying principles of quantum annealing, focusing on its ability to escape local minima and achieve a lower energy state, representing the optimal solution. The research aims to demonstrate the feasibility and potential advantages of employing quantum annealing in practical network routing applications, paving the way for more efficient and robust network designs.
The objective of this paper is to introduce the application of quantum annealing (QA) to electromagnetic (EM) engineering. We demonstrate that numerous EM design and inverse problems naturally admit a mathematical formulation in terms of binary quadratic interactions. By utilizing didactic, simplified examples, we illustrate how this underlying physical structure allows for a direct mapping of EM problems onto Quadratic Unconstrained Binary Optimization (QUBO) models and equivalent Ising Hamiltonians. All numerical experiments are conducted using Simulated Annealing as a classical proxy, since the primary contribution of this work is the QUBO formulation itself rather than its execution on quantum hardware. Our results suggest that a broad class of EM problems—including array thinning, reconfigurable intelligent surface optimization, subarray partitioning, and electromagnetic inverse scattering—are suitable candidates for optimization using near-future quantum annealing architectures, once hardware connectivity and noise floors reach the required specifications.