面向约束组合优化的可行张量网络快速设计
Quick design of feasible tensor networks for constrained combinatorial optimization
AI总结：
本文提出仅用初等数学快速设计可行张量网络的方法，无需惩罚函数即可处理约束组合优化，并在设施选址问题上验证了其构造速度与求解最优解的有效性。
摘要：
量子计算机有望实现大规模组合优化问题的快速求解。然而，其在保真度和量子比特数量方面的限制使其难以处理现实世界的问题。最近，一种使用张量网络的量子启发式求解器被提出，该求解器运行在经典计算机上。特别是，张量网络已被应用于实际场景中的约束组合优化问题。通过准备一个特定的张量网络来采样满足约束的状态，可以在不使用惩罚函数方法的情况下搜索可行解。以往的研究基于深刻的物理学知识，例如U(1)规范方案和高维晶格模型。在本研究中，我们设计了一种无需此类专门知识、仅使用初等数学即可设计可行张量网络的方法。一种方法是利用幂零矩阵操作构造张量网络。第二种方法是通过代数方式确定张量参数。我们从数学上证明，这种可行张量网络可以被构造以容纳各种类型的约束。为了进行原理验证，我们针对设施选址问题数值构造了一个可行张量网络，发现其构造速度远快于传统方法。随后，通过执行虚时间演化，始终能够获得可行解，并最终得到最优解。
英文摘要：
Quantum computers are expected to enable fast solving of large-scale combinatorial optimization problems. However, their limitations in fidelity and the number of qubits prevent them from handling real-world problems. Recently, a quantum-inspired solver using tensor networks has been proposed, which works on classical computers. Particularly, tensor networks have been applied to constrained combinatorial optimization problems for practical applications. By preparing a specific tensor network to sample states that satisfy constraints, feasible solutions can be searched for without the method of penalty functions. Previous studies have been based on profound physics, such as U(1) gauge schemes and high-dimensional lattice models. In this study, we devise to design feasible tensor networks using elementary mathematics without such a specific knowledge. One approach is to construct tensor networks with nilpotent-matrix manipulation. The second is to algebraically determine tensor parameters. We showed mathematically that such feasible tensor networks can be constructed to accommodate various types of constraints. For the principle verification, we numerically constructed a feasible tensor network for facility location problem, to find much faster construction than conventional methods. Then, by performing imaginary time evolution, feasible solutions were always obtained, ultimately leading to the optimal solution.
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