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 $A_i|ψ\rangle$ 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.
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⟩ 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 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.
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 H4 in STO-3G with CAS(4e,4o), the mapped sector agrees with independent determinant FCI to 3.1×10−15 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.
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.
We develop a sparse operator-centric realization of n-qubit variational quantum algorithms in the complex Clifford algebra 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-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 4.84×10−5, 2.19×10−3, and 3.67×10−3 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 1.3×10−12 for n≤6. 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.
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 CH4, 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 2n+1.
Quantum computing offers a promising paradigm for electromagnetic eigenmode analysis, enabling compact representations of complex field interactions and potential exponential speedup over classical numerical solvers. Recent efforts have applied variational quantum eigensolver (VQE) based methods to compute waveguide modes, demonstrating the feasibility of simulating TE and TM field distributions on noisy intermediate-scale quantum (NISQ) hardware. However, these studies typically employ manually designed, fixed-depth parameterized quantum circuits and uniform measurement-shot strategies, resulting in excessive quantum resource consumption, limited circuit expressivity, and reduced robustness under realistic noise conditions. To address these limitations, we propose an architecture and shot adaptive subspace variational quantum eigensolver for efficient microwave waveguide eigenmode simulation on NISQ devices. The proposed framework integrates a reinforcement learning (RL) based circuit design strategy and an adaptive shot allocation mechanism to jointly reduce quantum resource overhead. Specifically, the RL agent autonomously explores the quantum circuit space to generate hardware-efficient parameterized quantum circuits, while the adaptive measurement scheme allocates sampling resources according to Hamiltonian term weights. Numerical experiments on three- and five-qubit systems demonstrate that the proposed framework achieves accurate estimation of TE and TM mode eigenvalues, with a minimum absolute error down to 10−8 and reconstructed field distributions under noiseless conditions in excellent agreement with classical electromagnetic solutions.
Abstract The adaptive variational quantum eigensolver (ADAPT-VQE) is a promising approach for quantum simulation in the noisy intermediate-scale quantum era, offering advantages over traditional VQE methods by reducing circuit depth and mitigating challenges in classical optimisation. Although ADAPT-VQE can require fewer total measurements than fixed ansätze for matched precision, the combined cost of repeated energy and gradient evaluations still leads to a substantial measurement overhead in practice, motivating strategies to reduce it further. In this work, we study two complementary strategies to reduce this cost using only standard projective Pauli measurements. First, we apply variance-based shot allocation (variance-minimised shot assignment and variance-preserved shot reduction (VPSR)) to both Hamiltonian energy and operator-gradient measurements, extending techniques previously studied only for energy estimation in standard VQE. VPSR achieves shot reductions of up to 43% for H 2 and 51% for LiH relative to uniform allocation while maintaining chemical accuracy. Second, we introduce a Pauli-measurement reuse protocol that stores outcomes from VQE optimisation and reuses them for gradient evaluation in the next ADAPT iteration. Across six molecular benchmarks (4–14 qubits), measurement reuse yields only marginal savings confined to the smallest systems and is effectively negligible for larger sizes, pivot-based grouping alone provides the dominant reduction and should be considered the practical baseline. These results characterise the interplay between shot allocation, measurement grouping, and cross-iteration reuse, offering practical guidance on when each strategy is most beneficial.