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International Conference on Information Networking

Proper Cost Hamiltonian Design for Combinatorial Optimization Problems: A Boolean Function Approach

Jaeho Choi, Seunghyeok Oh, Soohyun Park, Jong‐Kook Kim, Joongheon Kim

Published Jan 13, 2021• 0 citations
Publisher

Abstract

Advanced researches on the variational quantum algorithms are actively conducted. In particular, the quantum approximate optimization algorithm (QAOA) is one of the promising variational quantum algorithms and can be applied to various graph-based problems, and is a promising algorithm that shows good performance even in small quantum computers. As is widely known, QAOA obtains the approximate solution via the expectation value of the cost Hamiltonian on the parameterized state. Therefore, in addition to finding the optimal parameters, the proper design of the cost Hamiltonian is important. This paper designs the cost function of the combinatorial optimization problem via Boolean function and maps it to the proper cost Hamiltonian. The proposed cost Hamiltonian design method is applied to the maximum independent set (MIS) and minimum dominating set (MDS) problems.

DOI
10.1109/icoin50884.2021.9333931

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Advanced researches on the variational quantum algorithms are actively conducted. In particular, the quantum approximate optimization algorithm (QAOA) is one of the promising variational quantum algorithms and can be applied to various graph-based problems, and is a promising algorithm that shows good performance even in small quantum computers. As is widely known, QAOA obtains the approximate solution via the expectation value of the cost Hamiltonian on the parameterized state. Therefore, in addition to finding the optimal parameters, the proper design of the cost Hamiltonian is important. This paper designs the cost function of the combinatorial optimization problem via Boolean function and maps it to the proper cost Hamiltonian. The proposed cost Hamiltonian design method is applied to the maximum independent set (MIS) and minimum dominating set (MDS) problems.
In the past few years, quantum computers and quantum algorithms have shown advantages compared to their classical counterparts. In combinatorial optimization problems, quantum approximate optimization algorithm(QAOA) has good performance. This paper focuses on the application of QAOA to one combinational problem called maximum independent set(MIS). Two different Hamiltonian construction methods are employed in this algorithm using IBM Qiskit. It is shown that the key is to design proper Hamiltonians to map the solution to the feasible states of the Hilbert space.
Advancements in Quantum Computing (QC) and Neural Combinatorial Optimization (NCO) represent promising steps in tackling complex computational challenges. On the one hand, Variational Quantum Algorithms such as QAOA can be used to solve a wide range of combinatorial optimization problems. On the other hand, the same class of problems can be solved by NCO, a method that has shown promising results, particularly since the introduction of Graph Neural Networks. Given recent advances in both research areas, we introduce Hamiltonian-based Quantum Reinforcement Learning (QRL), an approach at the intersection of QC and NCO. We model our ansatzes directly on the combinatorial optimization problem's Hamiltonian formulation, which allows us to apply our approach to a broad class of problems. Our ansatzes show favourable trainability properties when compared to the hardware efficient ansatzes, while also not being limited to graph-based problems, unlike previous approaches. In this work, we evaluate the performance of Hamiltonian-based QRL on a diverse set of combinatorial optimization problems to demonstrate the broad applicability of our approach and compare it to QAOA.
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal solution for graph coloring problems. However, this naive approach can increase the computation cost exponentially as the graph size and the number of colors increase. To mitigate such intolerable overhead, we investigate the methods to take the advantages of quantum computing properties to find a solution for multi-coloring graph problems in polynomial time. We utilize the variational quantum eigensolver (VQE) technique and quantum approximate optimization algorithm (QAOA) to find solutions for three combinatorial applications by both transferring each problem model to the corresponding Ising model and by using the calculated Hamiltonian matrices. Our results demonstrate that VQE and QAOA algorithms can find one of the best solutions for each application. Therefore, our modeling approach with hybrid quantum algorithms can be applicable for combinatorial problems in various fields to find an optimal solution in polynomial time.
Abstract Quantum approximate optimization algorithm (QAOA) is a promising framework for solving combinatorial optimization problems on near-term quantum devices. One such problem is the minimum dominating set (MDS), which is known to be NP-hard. Existing QAOA algorithms for this problem typically require numerous auxiliary qubits, increasing circuit overhead and hardware requirements. In this paper, we propose an auxiliary-qubit-free QAOA algorithm based on Hamiltonian evolution (AQFH-QAOA) for the MDS problem. Unlike previous studies that require numerous auxiliary qubits, our algorithm eliminates the need for auxiliary qubits, thereby significantly reducing circuit overhead. In addition, we present an auxiliary-qubit-free optimized implementation of the previously proposed Guerrero’s QAOA algorithm (AQFG-QAOA) by utilizing gate decomposition techniques. Through a detailed analysis of gate complexity, we evaluate the applicability of these two algorithms. Numerical experiments demonstrate that our proposed algorithm achieves competitive solution quality compared with existing QAOA algorithms, making it a promising candidate for implementation on near-term quantum devices.
Quantum computers are expected to accelerate solving combinatorial optimization problems, including algorithms such as Grover adaptive search and quantum approximate optimization algorithm (QAOA). However, many combinatorial optimization problems involve constraints which, when imposed as soft constraints in the cost function, can negatively impact the performance of the optimization algorithm. In this paper, we propose fermionic quantum approximate optimization algorithm (FQAOA) for solving combinatorial optimization problems with constraints. Specifically, FQAOA tackles the constrains issue by using fermion particle number preservation to intrinsically impose them throughout QAOA. We provide a systematic guideline for designing the driver Hamiltonian for a given problem Hamiltonian with constraints. The initial state can be chosen to be a superposition of states satisfying the constraint and the ground state of the driver Hamiltonian. This property is important since FQAOA reduced to quantum adiabatic computation in the large limit of circuit depth ppp and improved performance, even for shallow circuits with optimizing the parameters starting from the fixed-angle determined by Trotterized quantum adiabatic evolution. We perform an extensive numerical simulation and demonstrate that proposed FQAOA provides substantial performance advantages against existing approaches in portfolio optimization problems. Furthermore, the Hamiltonian design guideline is useful not only for QAOA but also Grover adaptive search and quantum phase estimation to solve combinatorial optimization problems with constraints. Since software tools for fermionic systems have been developed in quantum computational chemistry both for noisy intermediate-scale quantum computers and fault-tolerant quantum computers, FQAOA allows us to apply these tools for constrained combinatorial optimization problems.
Over the past few years, many researchers around the world have been keen to know the potential and efficiency of quantum computers. The researchers have focused on specific issues that classical computers cannot solve or issues that quantum computers can handle in a better way. Among these various attractive research topics in quantum computers, this paper introduces the Quantum Approximate Optimization Algorithm (QAOA) which guarantees relatively considerable performances in many combinatorial optimization problems. For the comprehensive understanding of QAOA, this paper also describes the approximate optimization, the Quantum Alternating Operator Ansatz, and applications. Besides the theories of QAOA and Quantum Alternating Operator Ansatz, this paper explains the applications of QAOA to major combinatorial optimization problems such as maximum cut (MaxCut) problem and the max-independent set (MIS) problem.