An Information-Minimal Geometry for Qubit-Efficient Optimization

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.