基于量子松弛的分支定界中的高效内部策略
Efficient Internal Strategies in Quantum Relaxation based Branch-and-Bound
AI总结：
本文将量子松弛QRAO引入分支定界框架提出QR-BnB方法，并开发基于泡利算符期望值的变量选择等内部策略，在最大割和旅行商问题上实现了更快的收敛并保持最优性。
摘要：
组合优化问题是在约束条件下寻找最优解，这是量子计算机的潜在应用之一。量子随机存取优化（QRAO）是一种量子优化算法，它将多个经典变量编码到单个量子比特中以构造量子哈密顿量，从而减少所需的量子比特数量。QRAO哈密顿量的基态能量为编码前原始问题的最优值提供了一个下界。这一性质使得QRAO哈密顿量可以作为原始问题的松弛，因此被称为量子松弛哈密顿量。在分支定界方法中，求解松弛问题起着重要作用。在本研究中，我们开发了基于量子松弛的分支定界（QR-BnB），这是一种将量子松弛纳入分支定界框架的方法。我们在实验中求解了最大割问题（MaxCut Problem）和旅行商问题（Travelling Salesman Problem）。在本研究的所有实例中，只要通过量子松弛成功计算出精确下界，我们就能获得最优解。松弛方法和变量选择等内部策略会影响分支定界的收敛性。因此，我们进一步为QR-BnB开发了内部策略，并研究了这些策略如何影响其收敛性。我们表明，通过泡利算符期望值进行的变量选择策略比朴素的随机选择具有更好的收敛性。QRAO仅处理无约束优化问题，但由于经典计算部分的分支定界过程，QR-BnB可以更灵活地处理约束条件。我们证明，在旅行商问题的实验中，通过利用约束中的信息，QR-BnB的收敛速度提高了三倍以上。
英文摘要：
A combinatorial optimization problem is to find an optimal solution under the constraints. This is one of the potential applications for quantum computers. Quantum Random Access Optimization (QRAO) is the quantum optimization algorithm that encodes multiple classical variables into a single qubit to construct a quantum Hamiltonian, thereby reducing the number of qubits required. The ground energy of the QRAO Hamiltonian provides a lower bound on the original problem's optimal value before encoding. This property allows the QRAO Hamiltonian to be used as a relaxation of the original problem, and it is thus referred to as a quantum relaxed Hamiltonian. In the Branch-and-Bound method, solving the relaxation problem plays a significant role. In this study, we developed Quantum Relaxation based Branch-and-Bound (QR-BnB), a method incorporating quantum relaxation into the Branch-and-Bound framework. We solved the MaxCut Problem and the Travelling Salesman Problem in our experiments. In all instances in this study, we obtained the optimal solution whenever we successfully computed the exact lower bound through quantum relaxation. Internal strategies, such as relaxation methods and variable selection, influence the convergence of the Branch-and-Bound. Thus, we have further developed the internal strategies for QR-BnB and examined how these strategies influence its convergence. We show that our variable selection strategy via the expectation value of the Pauli operators gives better convergence than the naive random choice. QRAO deals with only unconstrained optimization problems, but QR-BnB can handle constraints more flexibly because of the Branch-and-Bound processes on the classical computing part. We demonstrate that in our experiments with the Travelling Salesman Problem, the convergence of QR-BnB became more than three times faster by using the information in the constraints.
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