一种基于变分量子本征求解器的半定规划问题割平面框架
A variational quantum eigensolver-based cutting plane framework for semidefinite programming problems
- Bilkent University（比尔肯特大学）
- Sabancı University（萨贝里大学）
- Rensselaer Polytechnic Institute（伦斯勒理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出基于变分量子本征求解器的半定规划割平面框架，在SDPLIB控制族数据集上，其最优性间隙闭合率达32%-82%，揭示了混合量子-经典方法的相关效应。
AI中文摘要：
半定规划在优化领域发挥关键作用，对控制理论、机器学习和组合优化具有广泛影响。尽管半定规划属于多项式可解问题，但若干常用算法依赖于线性代数步骤，其运行时间随矩阵维度呈三次方增长，且需要将矩阵本身存储在内存中，内存开销为二次方级别。本研究中，我们提出用变分量子本征求解器（Variational Quantum Eigensolver, VQE）替代该线性代数步骤，其量子比特需求随矩阵维度呈对数增长，并首次在割平面框架内实现了此类方法的端到端实现，同时提出一种基于算子的本征态（ansatz），其纠缠结构直接从候选矩阵的泡利支撑中读取。在SDPLIB的控制族数据集上，与由精确本征分解驱动的相同方案相比，变分神谕全程生成有效割平面，初始最优性间隙的闭合率为32%至82%，而精确神谕的闭合率接近恒定的75%至82%。对该方法进行端到端实现与测量揭示了若干仅通过理论分析无法观察到的效应：实际内存消耗位置、将矩阵适配到量子寄存器所需的填充如何误导变分优化器，以及为何候选矩阵在泡利基下呈稠密性，使基于算子的本征态退化为全纠缠。我们报告这些发现，并讨论其对近期混合量子-经典方法的启示。
英文摘要：
Semidefinite programming plays a key role in optimization, with broad impact across control theory, machine learning, and combinatorial optimization. Although semidefinite programs are polynomially solvable, several commonly used algorithms rest on a linear-algebraic step whose running time grows cubically with the matrix dimension and which requires the matrix itself to be held in memory, at quadratic cost. In this study, we propose replacing it with a variational quantum eigensolver, whose qubit requirement is logarithmic in the matrix dimension, and present the first end-to-end implementation of such an approach within a cutting-plane framework, together with an operator-derived ansatz whose entanglement structure is read directly from the Pauli support of the candidate matrix. Evaluated on the control family of SDPLIB against an identical scheme driven by an exact eigendecomposition, the variational oracle produces valid cuts throughout, closing 32 to 82% of the initial optimality gap against a near-constant 75 to 82% for the exact oracle. Implementing and measuring the method end to end surfaces several effects not visible from theoretical analyses alone: where memory is actually consumed, how the padding required to fit a matrix onto a quantum register can mislead the variational optimizer, and why the candidate matrices prove dense in the Pauli basis, reducing the operator-derived ansatz to full entanglement. We report these findings and discuss their implications for near-term hybrid quantum-classical approaches.
