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arXiv:2607.16869

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

Federico Zahariev, Vanda Glezakou

quant-phphysics.chem-phSubmitted Jul 18, 2026• Announced Jul 21, 2026• 0 citations• Revised Oct 7, 2026 (v2)
v1 · Jul 18, 2026v2 · Oct 7, 2026
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Abstract

One should not assume that reconstructing an electronic observable with a smaller set of measurement settings will cut the number of repetitions in proportion. Each setting is simply a basis change and qubit readout, but the savings are not guaranteed to be commensurate. Take two electrons of opposite spin in two spatial orbitals: the orbital degree of freedom of each electron acts as a logical qubit and rotations are independent in the spin sectors. From the observable's one- and two-qubit coefficients we derive an exact floor for the setting count and a decomposition to meet it in no more than three settings. With fewer settings, the minimum error in the two-qubit correlation coefficients is set by the singular values left out. A case in point is an exchange with equal nonzero couplings on the three Pauli axes. This demands three orbital settings yet only one joint measurement. We offer a Clifford circuit for four physical qubits to effect the necessary mapping of Pauli products under conjugation. In the case of a one-parameter family of logical singlet-triplet mixtures, the orbital-to-Clifford shot cost ratio is three at maximum mixing and trends toward 3/2 near the logical singlet. Where spin sectors may be mixed, isotropic exchange needs one orbital setting. For spin-conserving Hamiltonians with $N$ spatial orbitals encoded via Jordan-Wigner, the $X$-rank, the binary rank of the $X$ parts of its strings, of any commuting Pauli string family is no more than $2(N-1)$. Searches with the Robust Adaptive Nature-inspired Global Explorer (RANGE) reach this limit for CH$_4$, with the rank of the Hamiltonian support vouching for the maximality of the search values. And for all nonidentity Pauli strings on $n$ qubits, semidefinite witnesses from the Lovász theta number give a lower bound on the Clifford setting count that agrees with the integer cover and counting bounds at $2^n+1$.

Comments
Revised version. Heavy-element X-ranks recomputed from audited operators, text revised throughout, references updated. 21 pages, 2 figures

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