Beyond Orbital Rotations: Correlation-Rank Limits and Clifford-Accessible Measurement, from Algebra and Global Optimization

Abstract

Algebra and RANGE global optimization play complementary, explicitly separated roles in identifying measurement structure beyond orbital rotations. In the fixed (1,1)-particle sector of two spatial orbitals per spin, algebra proves that one particle-number-preserving orbital-rotation context contributes a rank-one two-body correlation block $T$: an observable needs at least $\mathrm{rank}\,T$ such contexts, and its best $K$-context correlation-block approximation is exactly the Eckart-Young singular-value tail, attained by the truncated SVD. A continuous RANGE search over the physical rotation angles independently corroborates this exact trade-off. A Bell-diagonal, Heisenberg-type witness has correlation rank three: it needs at least three orbital-rotation contexts, while one explicit physical Clifford circuit measures its commuting Pauli representatives. For spin-conserving Jordan-Wigner molecular Hamiltonians we also prove the parity ceiling $r_X \le 2(N-1)$ for any Pauli subset, tight even within commuting subsets; $X$-rank is a routing diagnostic, and the strict separation is carried by the correlation-rank theorem. The discrete mode of RANGE locates high-$X$-rank commuting families across molecular and production f-element Hamiltonians, finding ceiling-saturating witnesses for CH4 and NdO; values are best found unless a proved ceiling is attained. Applying the companion certificate framework, enlarging product settings by fully commuting, Clifford-accessible settings reduces the certified leading shot cost by 31-70% on four 29-35-qubit f-element Hamiltonians, a QWC-versus-QWC+FC result rather than a Gaussian-versus-Clifford pricing. Controlled-Pauli insertions in Hadamard tests are Clifford; these zero-$T$ statements concern measurement circuitry only, while shot counts and state preparation retain their full costs.