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arXiv:2608.00560

Adaptive operator-generated subspaces for effective many-body Hamiltonians

Ginanjar Utama, Hermawan Kresno Dipojono

quant-phphysics.comp-phSubmitted Aug 1, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

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.

Comments
13 pages, 4 figures, 8 tables

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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 Ai∣ψ⟩A_i|ψ\rangleAi​∣ψ⟩ 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 H4_44​ 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.
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We develop a sparse operator-centric realization of nnn-qubit variational quantum algorithms in the complex Clifford algebra Cl(2n,C)≅M(2n,C)\mathrm{Cl}(2n,\mathbb{C}) \cong M(2^n,\mathbb{C})Cl(2n,C)≅M(2n,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-YYY 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 YYY factors has zero ADAPT gradient, while odd-YYY rotations preserve the real sector. For the critical open transverse-field Ising chain, a depth-three Hamiltonian variational ansatz gives relative energy errors 4.84×10−54.84\times10^{-5}4.84×10−5, 2.19×10−32.19\times10^{-3}2.19×10−3, and 3.67×10−33.67\times10^{-3}3.67×10−3 for n=4,5,6n=4,5,6n=4,5,6. A compact local ADAPT pool is exact at n=4n=4n=4 but leaves residual errors at larger sizes; a systematic contiguous three-local odd-YYY pool reaches relative errors below 1.3×10−121.3\times10^{-12}1.3×10−12 for n≤6n\leq6n≤6. In 100-seed finite-shot tests at n=4n=4n=4, fixed-shot selection succeeds in 0/1000/1000/100 runs, whereas uniform escalation and confidence-bound racing each succeed in 84/10084/10084/100 runs; racing lowers median shots by 34%34\%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.
clifford_qc is a Python research toolkit for constructing quantum models, comparing variational and subspace eigensolvers, and estimating measurement resources. A common sparse Pauli algebra connects Hamiltonians, density operators, gates, and candidate-selection observables; dedicated interfaces handle circuits, execution, and statistical estimates. A worked gradient example illustrates shared measurements, and a molecular workflow separates reusable preparation, solving, and reference validation. Matrix-element banks support virtual subspaces, exact Ritz residuals, and folded-spectrum roots. Process-isolated benchmarks show that packed coefficients and streaming reduce observed peak memory on the larger tested inputs at increased runtime. Measurement studies separate setting counts, shot requirements, and device cost: fully commuting readout certifies a fixed first-order Ritz functional with fewer shots, while anticommuting cliques help some fixed reference energies and not others. Entangling-gate time and connectivity can offset shot savings or make a protocol inadmissible under illustrative device models. Further studies expose subspace bias, indeterminate encoding comparisons, failed approximate-restriction screens, and high residual-measurement costs under the tested uniform allocations. Generated tables, provenance records, a versioned evidence contract, and manuscript checks connect claims to their inputs. These small-system results support reproducible method development and establish neither hardware performance nor a general simulation speedup.
Adaptive Derivative-Assembled Problem-Tailored variational quantum eigensolvers (ADAPT-VQE) represent one of the most promising approaches for quantum chemistry on near-term quantum devices. However, their optimization is slow and may stall due to vanishing parameters and redundant operators in the ansatz. In this work, we propose a simple strategy of operator elimination that removes non-contributing operators from the pool once they are detected, enabling the optimization to continue progressing toward convergence. We examine two variants, with and without pool restoration after elimination, and find that the former converges more smoothly and faster than the latter and the standard ADAPT-VQE. To capture dynamical correlations between the active space and its environment, we combine ADAPT-VQE with our recently developed downfolding approach, the one-body downfolding framework (OBDF). In OBDF, the bare molecular Hamiltonian in the active space is replaced by a correlated effective Hamiltonian that incorporates dynamical correlation effects outside the active space. We benchmark our implementation on a linear \ce{H_6} chain, an \ce{H_6} lattice, an \ce{H_6} ring, and the \ce{N_2} molecule using the OpenFermion simulator. Our results show that operator elimination significantly reduces circuit depth and iteration count, and that OBDF-ADAPT-VQE yields energies closer to the full configuration interaction (FCI) reference than the standard approach within the same active space.
Circuit depth and measurement count are the main limitations for chemical applications of the leading variational quantum algorithms on quantum computers. We show that both resource requirements can be reduced by augmenting adaptive variational quantum algorithms with post-optimization corrective techniques. We focus on two such techniques---Hamiltonian-moment Krylov subspace diagonalization and the identity-augmented quantum self-consistent equation of motion (q-sc-EOM)---applied to optimized ADAPT-VQE states without further ansatz growth or nonlinear optimization. Across molecular systems studied with up to 20 qubits, first-order Krylov correction reduces the optimized iterations (and the corresponding number of adaptive layers) needed to reach the 1.6 mEh1.6~\mathrm{m}E_h1.6 mEh​ threshold by up to approximately sixfold, with a general trend toward larger reductions at higher determinant Rényi entropy. We also find measurement-cost reductions of up to approximately fortyfold using a suitable resource-cost model. Identity-augmented q-sc-EOM improves molecular potential-energy curves obtained from truncated ADAPT-VQE ansätze with limited expressivity and recovers accuracy from states produced by sampled operator ranking without additional gradient-based operator selection, providing an advantage when small pool gradients make selection measurement intensive. Together, these post-optimization corrections offer routes to improve energy accuracy and reduce adaptive circuit growth in quantum algorithms as quantum hardware advances.