Mixed discrete-continuous optimization is central to engineering design, where discrete choices interact with continuous fields. These problems are difficult due to high-dimensional, complex search spaces. To tackle them, Quantum Annealing (QA) is promising, yet its native binary nature supports only discrete variables, making accurate and efficient encodings of continuous quantities a central challenge. Existing approaches either split the coupled problem, mapping discrete decisions to QA while solving continuous fields classically, or use fixed-bit-depth encodings. The former compromises QA's global search advantages; the latter can underrepresent dynamic range or inflate the number of binary variables. We show that simply increasing bit depth can even degrade performance on current QA hardware, underscoring the need for alternative encodings. In response, we introduce an adaptive encoding strategy for continuous variables in QA that enables efficient treatment of coupled mixed-variable problems. We propose an update strategy for the representable ranges of the continuous variables and demonstrate its utility by integrating it into the minimum complementary energy formulation for structural design optimization, which provides a single, coupled constrained problem. We apply a quadratic penalty method where we update the representation of the continuous variables while targeting the full original objective, preserving QA's global search capability. On a published benchmark, the size optimization of a composite rod, our adaptive encoding improves solution quality under a fixed binary variable budget, demonstrating a superior precision-resource trade-off. Since the framework generalizes beyond structural design, it offers practical guidance for encoding continuous variables for QA and indicates that adaptive representations can enhance precision on current hardware.

翻译：混合离散-连续优化是工程设计的核心问题，其中离散选择与连续场相互耦合。由于高维复杂的搜索空间，这类问题求解难度较大。量子退火（QA）为解决此类问题提供了前景，但其原生二进制特性仅支持离散变量，因此连续量的精确高效编码成为关键挑战。现有方法要么将耦合问题拆分——将离散决策映射至QA而经典求解连续场，要么采用固定位深编码。前者会削弱QA的全局搜索优势；后者可能无法充分表征动态范围或导致二进制变量数量膨胀。我们发现，在当前QA硬件上单纯增加位深甚至可能降低性能，这凸显了替代编码方案的必要性。为此，我们提出一种用于QA中连续变量的自适应编码策略，能够高效处理耦合的混合变量问题。我们提出了连续变量可表示范围的更新策略，并通过将其集成至结构设计优化中的最小余能公式（该公式提供了单一耦合约束问题）来验证其有效性。我们采用二次罚函数法，在保持完整原始目标函数的同时更新连续变量的表示，从而保留QA的全局搜索能力。在已发布的复合材料杆尺寸优化基准测试中，我们的自适应编码在固定二进制变量预算下提升了求解质量，展现出更优的精度-资源权衡。由于该框架可推广至结构设计之外的领域，它为QA中连续变量的编码提供了实用指导，并表明自适应表示能够提升当前硬件上的求解精度。