通过提取半对称性来降低QUBO密度
Reducing QUBO Density by Factoring Out Semi-Symmetries
AI总结：
本文提出通过识别并提取QUBO矩阵中的半对称性至辅助量子比特来降低矩阵密度，理论上保持能谱不变，实验证明该方法能将QAOA电路深度和耦合数减少45%，并优化量子退火性能。
摘要：
量子近似优化算法（QAOA）和量子退火是解决组合优化问题（如表示为二次无约束二值优化（QUBO）的问题）的突出方法。这些算法旨在最小化目标函数 $x^T Q x$，其中 $Q$ 是 QUBO 矩阵。然而，QAOA 电路中双量子比特 CNOT 门的数量和量子退火中问题嵌入的复杂度与 $Q$ 中非零耦合的数量呈线性关系，导致了重大的计算和错误相关挑战。为了解决这个问题，我们引入了 QUBO 矩阵中半对称性的概念，并提出了一种识别这些对称性并将其提取到辅助量子比特上的算法。半对称性频繁出现在优化问题中，如最大团、哈密顿回路、图着色和图同构。我们从理论上证明，修改后的 QUBO 矩阵 $Q_{\ ext{mod}}$ 保留了与原始 $Q$ 相同的能谱。在上述问题上的实验评估表明，我们的算法将耦合数量和 QAOA 电路深度最多减少了 45%。对于量子退火，这些减少也导致了更稀疏的问题嵌入、更短的量子比特链和更好的性能。这项工作突出了利用 QUBO 矩阵结构来优化量子算法的效用，推进了它们对现实世界组合优化问题的可扩展性和实际适用性。
英文摘要：
Quantum Approximate Optimization Algorithm (QAOA) and Quantum Annealing are prominent approaches for solving combinatorial optimization problems, such as those formulated as Quadratic Unconstrained Binary Optimization (QUBO). These algorithms aim to minimize the objective function $x^T Q x$, where $Q$ is a QUBO matrix. However, the number of two-qubit CNOT gates in QAOA circuits and the complexity of problem embeddings in Quantum Annealing scale linearly with the number of non-zero couplings in $Q$, contributing to significant computational and error-related challenges. To address this, we introduce the concept of \textit{semi-symmetries} in QUBO matrices and propose an algorithm for identifying and factoring these symmetries into ancilla qubits. \textit{Semi-symmetries} frequently arise in optimization problems such as \textit{Maximum Clique}, \textit{Hamilton Cycles}, \textit{Graph Coloring}, and \textit{Graph Isomorphism}. We theoretically demonstrate that the modified QUBO matrix $Q_{\text{mod}}$ retains the same energy spectrum as the original $Q$. Experimental evaluations on the aforementioned problems show that our algorithm reduces the number of couplings and QAOA circuit depth by up to $45\%$. For Quantum Annealing, these reductions also lead to sparser problem embeddings, shorter qubit chains and better performance. This work highlights the utility of exploiting QUBO matrix structure to optimize quantum algorithms, advancing their scalability and practical applicability to real-world combinatorial problems.
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