An Information-Minimal Geometry for Qubit-Efficient Optimization

Qubit-efficient optimization seeks to represent an $N$-variable combinatorial problem within a Hilbert space smaller than $2^N$, using only as much quantum structure as the objective itself requires. Quadratic unconstrained binary optimization (QUBO) problems, for example, depend only on pairwise information -- expectations and correlations between binary variables -- yet standard quantum circuits explore exponentially large state spaces. We recast qubit-efficient optimization as a geometry problem: the minimal representation should match the $O(N^2)$ structure of quadratic objectives. The key insight is that the local-consistency problem -- ensuring that pairwise marginals correspond to a realizable global distribution -- coincides exactly with the Sherali-Adams level-2 polytope $\mathrm{SA}(2)$, the tightest convex relaxation expressible at the two-body level. Previous qubit-efficient approaches enforced this consistency only implicitly. Here we make it explicit: (a) anchoring learning to the $\mathrm{SA}(2)$ geometry, (b) projecting via a differentiable iterative-proportional-fitting (IPF) step, and (c) decoding through a maximum-entropy Gibbs sampler. This yields a logarithmic-width pipeline ($2\lceil\log_2 N\rceil + 2$ qubits) that is classically simulable yet achieves strong empirical performance. On Gset Max-Cut instances (N=800--2000), depth-2--3 circuits reach near-optimal ratios ($r^* \approx 0.99$), surpassing direct $\mathrm{SA}(2)$ baselines. The framework resolves the local-consistency gap by giving it a concrete convex geometry and a minimal differentiable projection, establishing a clean polyhedral baseline. Extending beyond $\mathrm{SA}(2)$ naturally leads to spectrahedral geometries, where curvature encodes global coherence and genuine quantum structure becomes necessary.