Operator-centric Clifford algebra for variational eigensolvers and finite-shot adaptive selection

We develop a sparse operator-centric realization of $n$-qubit variational quantum algorithms in the complex Clifford algebra $\mathrm{C}\mathrm{l}(2n,\mathbb{C})\cong M(2^n,\mathbb{C})$. Density operators, gates, observables, channels, fermionic modes, and adaptive-selection observables are represented in one Pauli-word algebra, with the Jordan–Wigner map providing the exact bridge to anticommuting Clifford generators. We distinguish general Pauli-word rotations from Spin-group rotors and derive an exact transpose-parity rule: for real Hamiltonians and real states, every candidate Pauli word containing an even number of $Y$ factors has zero ADAPT gradient, while odd-$Y$ rotations preserve the real sector. For the critical open transverse-field Ising chain, a depth-three Hamiltonian variational ansatz gives relative energy errors $4.84\times {10}^{-5}$, $2.19\times {10}^{-3}$, and $3.67\times {10}^{-3}$ for $n=4,5,6$. A compact local ADAPT pool is exact at $n=4$ but leaves residual errors at larger sizes; a systematic contiguous three-local odd-$Y$ pool reaches relative errors below $1.3\times {10}^{-12}$ for $n\le 6$. In 100-seed finite-shot tests at $n=4$, fixed-shot selection succeeds in $0\mathrm{/}100$ runs, whereas uniform escalation and confidence-bound racing each succeed in $84\mathrm{/}100$ runs; racing lowers median shots by $34\mathrm{\%}$. We claim no asymptotic speedup over matrix methods. The contribution is a corrected algebraic formulation, an exact pool-pruning rule, and a reproducible study of measurement-limited adaptive selection.