Electronic-structure and embedding workflows terminate in effective many-body Hamiltonians, whereas quantum eigensolver studies often start from hand-built qubit models. We present the Adaptive Clifford-Algebra Subspace Eigensolver (A-CASE), a single-reference, operator-generated Rayleigh--Ritz method. Overlap, Hamiltonian, observable, and response matrices are reconstructed from one shared Pauli-expectation bank, while adaptive growth scores overlap-aware local pencils and rejects symmetry leakage or near-linear dependence. A strict FCIDUMP adapter supplies the active-space boundary. For linear H$_4$ in STO-3G with CAS(4e,4o), the mapped sector agrees with independent determinant FCI to $3.1\times10^{-15}$ Ha. At a nine-vector budget A-CASE has a $3.019$ mHa error; replacing the determinant reference by a two-operator ADAPT-VQE state reduces it to $0.342$ mHa, without implying a matched total-cost advantage. Under a matched contract, the fixed-reference route uses one state preparation versus ADAPT-VQE's ninety but measures roughly an order of magnitude more Pauli words. Exact fixed-angle ADAPT-GCIM gives $13.364$ mHa at the nearest size match and $10.674$ mHa at the iteration match, with its transition-pair burden reported separately. Across a broader benchmark ladder, fixed Krylov bases are generally more accurate and often narrower but substantially less well conditioned. A grouped bootstrap propagates finite-shot variability through thresholding, diagonalization, root matching, spectral weights, susceptibility, and broadening; its bands are explicitly heuristic and conditional, not finite-sample confidence certificates. The work establishes an executable path from an interchange Hamiltonian to energies, correlations, and response, without claiming materials accuracy, favorable scaling, or quantum advantage.
We develop a sparse operator-centric realization of $n$-qubit variational quantum algorithms in the complex Clifford algebra $\mathrm{Cl}(2n,\mathbb{C}) \cong M(2^n,\mathbb{C})$. 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 formulate the familiar odd-$Y$ restriction for real-state adaptive ansatzes as an exact transpose-parity statement: 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 $4.84\times10^{-5}$, $2.19\times10^{-3}$, and $3.67\times10^{-3}$ 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 $1.3\times10^{-12}$ for $n\leq6$. In 100-seed finite-shot tests at $n=4$, fixed-shot selection succeeds in $0/100$ runs, whereas uniform escalation and confidence-bound racing each succeed in $84/100$ runs; racing lowers median shots by $34\%$. We claim no asymptotic speedup over matrix methods. The contribution is a corrected algebraic formulation, a density-operator derivation and implementation of the real-sector pool filter, and a reproducible study of measurement-limited adaptive selection.
Quantum subspace methods are often compared by basis dimension or energyerror, although their dominant experimental costs arise from different statepreparations, measurement settings, and shot allocations. We present theDyadic Adaptive Clifford-Algebra Subspace Eigensolver (DA-CASE), whose basisstates are virtual directions $A_i|\psi\rangle$ generated from one reference.Overlap, Hamiltonian, and observable matrices are reconstructed from onecached set of Pauli expectations on that reference. The method thereforetrades multiple prepared basis states for a potentially wide measurement bank.We make that trade explicit on a frozen eight-qubit H$_4$ Hamiltonian. Twogenerator resolutions reach the same nine-dimensional subspace and the sameenergy to machine precision, while the retained bank changes from 7371 to 2240Pauli words. A reference-conditioned symmetry test certifies the narrower spanwithout asserting that its individual Pauli words conserve the sector asabstract operators. Independently, a dyadic commuting hierarchy reduces thedeterminant bank from 913 qubit-wise-commuting settings to 64 fully commutingsettings, while exposing the added logical-CX cost. In a separate four-qubitfinite-shot diagnostic, covariance-aware allocation reduces theprojected-matrix variance target by 68.9%. Mode-wise overlap regularizationremoves the observed catastrophic energy estimates and lowers RMSE, butdoubles the median error relative to a fixed cutoff. These are small-instanceexact and Monte Carlo results, not a hardware demonstration, scaling result,or quantum advantage claim. The contribution is a single-referencemeasurement architecture and a resource ledger that keeps contexts, settings,shots, circuit depth, and post-selection retries in their proper units.
Ginanjar Utama, H. K. Dipojono· 0 citations
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