Qubit-efficient optimization studies how large combinatorial problems can be addressed with quantum circuits whose width is far smaller than the number of logical variables. In quadratic unconstrained binary optimization (QUBO), objective values depend only on one- and two-body statistics, yet standard variational algorithms explore exponentially large Hilbert spaces. We recast qubit-efficient optimization as a geometric question: what is the minimal representation the objective itself requires? Focusing on QUBO problems, we show that enforcing mutual consistency among pairwise statistics defines a convex body -- the level-2 Sherali-Adams polytope -- that captures the information on which quadratic objectives depend. We operationalize this geometry in a minimal variational pipeline that separates representation, consistency, and decoding: a logarithmic-width circuit produces pairwise moments, a differentiable information projection enforces local feasibility, and a maximum-entropy ensemble provides a principled global decoder. This information-minimal construction achieves near-optimal approximation ratios on large unweighted Max-Cut instances (up to N=2000) at shallow depth, indicating that pairwise polyhedral geometry already captures the relevant structure in this regime. By making the information-minimal geometry explicit, this work establishes a clean baseline for qubit-efficient optimization and sharpens the question of where genuinely quantum structure becomes necessary.

翻译：量子比特高效优化研究如何用量子电路解决大规模组合问题，这些电路的宽度远小于逻辑变量的数量。在二次无约束二进制优化（QUBO）中，目标函数值仅取决于单体和二体统计量，然而标准的变分算法探索的是指数级大的希尔伯特空间。我们将量子比特高效优化重新表述为一个几何问题：目标函数本身所需的最小表示是什么？聚焦于QUBO问题，我们证明了强制执行成对统计量之间的相互一致性定义了一个凸体——即二阶Sherali-Adams多面体——它捕获了二次目标函数所依赖的信息。我们将此几何结构操作化为一个最小变分流程，该流程分离了表示、一致性和解码：一个对数宽度的电路生成成对矩，一个可微分的信息投影强制执行局部可行性，而一个最大熵系综提供了一个有原则的全局解码器。这种信息最小构造在浅层深度下，于大规模无权重Max-Cut实例（高达N=2000）上实现了接近最优的近似比，表明在此体系中成对多面体几何已经捕获了相关结构。通过使信息最小几何显式化，这项工作为量子比特高效优化建立了一个清晰的基线，并尖锐地提出了在何处真正需要量子结构的问题。