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SPIE Photomask Technology Conference

A formulation of mask optimization into QUBO model for Ising machines

Yukihide Kohira, Haruki Nakayama, Naoki Nonaka, Tomomi Matsui, Atsushi Takahashi, Chikaaki Kodama

Published Nov 21, 2023• 0 citations
Publisher

Abstract

Recently, Ising machines, which solve Quadratic Unconstrained Binary Optimization (QUBO) problems in a short computation time, have attracted attention. In this paper, we propose a mask optimization method using an Ising machine to obtain a mask with high fidelity to target patterns and high tolerance to process variation. In the proposed method, a mask pattern is improved by repeatedly solving QUBO problems by the Ising machine. In experiments, we applied the proposed method using the Ising machine to various patterns and evaluated it in comparison with existing methods in fidelity to target patterns, tolerance to process variation, and execution time.

DOI
10.1117/12.2687615

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Recently, Ising machines, which solve Quadratic Unconstrained Binary Optimization (QUBO) problems in a short computation time, have attracted attention. In this paper, we propose a mask optimization method using an Ising machine to obtain a mask with high fidelity to target patterns and high tolerance to process variation. In the proposed method, a mask pattern is improved by repeatedly solving QUBO problems by the Ising machine. In experiments, we applied the proposed method using the Ising machine to various patterns and evaluated it in comparison with existing methods in fidelity to target patterns, tolerance to process variation, and execution time.
In the field of microlithography, conventional computers are widely used for mask optimization. Recent progress of quantum and quantum-inspired computers has encouraged the development of quantum algorithms for numerous applications. So far, no method has been established for solving mask optimization problems with quantum computers. We introduced a simple model that describes the mask optimization problem as a quadratic unconstrained binary optimization (QUBO) problem, which is easily implemented on these computers. For simplicity, we assume there exists a target image profile on the wafer. The target can be the image of an existing mask or a virtual ideal mask which may be designed as a pixelated mask having a continuous transmission distribution. The solution is evaluated as the difference between the simulated image profile on the wafer surface and the target profile.
Ising machines have attracted much attention in recent years because of their potential in solving combinatorial optimization problems that are difficult to be solved using conventional computers. Ising machines solve the optimization problem that can be transformed into a quadratic unconstrained binary optimization (QUBO) problem. In the case of a problem with constraints, named a constraint problem, an objective function and constraint function with penalty coefficients are combined into a single Hamiltonian of a QUBO problem. When solving a constraint problem, the accuracy of the solution decreases compared to a QUBO problem with an objective function only. This is because penalty coefficients are set to excessively large values to prevent constraint violations. As a result, it becomes difficult to minimize the objective function. To solve this problem, this paper proposes a method that partitions constraint functions and reduces the value of the penalty coefficients. The performance evaluation using the traveling salesperson problem (TSP) with one-hot constraints demonstrates that the proposed method can improve the solution accuracy compared to the conventional method.
Ising machines, including quantum annealing machines, can efficiently solve combinatorial optimization problems by formulating them as Quadratic Unconstrained Binary Optimization (QUBO). However, it is known that the solving performance of Ising machines deteriorates as the energy landscape represented by the QUBObecomes more complex. Simplifying the coefficients of QUBOmatrix can improve the performance of solving combinatorial optimization problems using an Ising machine. In this paper, we propose QUBO simplification methods for Ising machines using singular value decomposition (SVD) and coefficient elimination based on their absolute value (ABS). The proposed SVD method simplifies QUBO by capturing the global structure of the QUBO matrix. QUBO is further simplified by combining it with the simple ABS method. The simplified QUBO is expected to have a less complex energy landscape, while maintaining the global structure of the original, enabling the Ising machine to efficiently find the (semi-)ground state.We demonstrate the numerical experiments using an Ising machine to evaluate the effectiveness of the SVD method alone, the ABS method alone, and the two methods combined.
Abstract 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.
Ising machines have recently gained a significant attention as a promising approach to combinatorial optimization problems, which are notoriously challenging for conventional computers to solve efficiently. In contrast to conventional computers, Ising machines operate based on the principles of the Ising model derived from statistical mechanics. Ising machines tackle combinatorial optimization problems by transforming them into Quadratic Unconstrained Binary Optimization (QUBO) formulations. The Traveling Salesperson Problem (TSP) is a prominent example of such optimization challenges and is widely recognized for its computational difficulty on conventional computers. Although Ising machines offer the potential for efficiently solving TSPs, the process of a QUBO formulation, namely the generation of a QUBO matrix that represents a TSP, demands substantial memory resources and considerable computation overheads. This study proposes a compressed QUBO format for TSP. This format reduces the dimensionality of the QUBO matrix for TSP by leveraging repetitive values, structural patterns, and sparsity within the matrix. The proposed format enhances the efficiency of data preprocessing, significantly reducing both the computational time required for QUBO matrix generation and the associated memory usage. These improvements contribute to the practical applicability of digital hardware-based Ising machines for addressing large-scale combinatorial optimization problems, thereby expanding their utility in solving complex real-world challenges.
With the increasing emphasis on energy efficiency and sustainability, optimizing the energy supply of base stations, the major energy consumers in communication networks, has become a critical problem. In addition, for real-time and scalable energy optimization in sustainable base station systems, it is necessary to obtain highly-efficient solutions in a shorter time. We formulate the base station power optimization problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem, where the goal is to minimize a combination of power costs while satisfying operational constraints. Furthermore, to improve computational efficiency, we propose a QUBO simplification method that cuts down QUBO coefficients with small absolute values to zero. We solve both the original and simplified QUBOs using an Ising machine. Experimental results show that our QUBO simplification approach accelerates computation by up to 10 X on large-scale instances, with minimal degradation in solution quality.