Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.17443

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

Ginanjar Utama, Hermawan Kresno Dipojono

quant-phSubmitted Jul 20, 2026• Announced Jul 21, 2026• 0 citations• Revised Jul 31, 2026 (v2)
v1 · Jul 20, 2026v2 · Jul 31, 2026
arXivHTMLPDF

Abstract

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.

Comments
9 pages plus 3 pages Supplemental Material; 6 figures

Related papers

clifford_qc: A Python Toolkit for Quantum Simulation

Ginanjar Utama, Hermawan Kresno Dipojono

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.

quant-pharXiv:2609.23059•Sep 22, 2026•0 citations

Adaptive operator-generated subspaces for effective many-body Hamiltonians

Ginanjar Utama, Hermawan Kresno Dipojono

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.

quant-pharXiv:2608.00560•Aug 4, 2026•0 citations

DA-CASE: reusable measurements for adaptive quantum subspaces

Ginanjar Utama, Hermawan Kresno Dipojono

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.

quant-pharXiv:2608.08739•Aug 11, 2026•0 citations

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

Federico Zahariev, Vanda Glezakou

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 $2^n+1$.

quant-pharXiv:2607.16869•Jul 21, 2026•0 citations

Investigating Interacting Fermionic Models with Locality-Preserving Qubit Encodings

Ashutosh P. Tripathi, Debasish Banerjee, Sandip Maiti +1 more

We investigate the utility of the locality-preserving Derby-Klassen (DK) fermion-to-qubit mapping [arXiv:2003.06939] for variational quantum simulation of two-dimensional $t$-$V$ and Fermi-Hubbard models. The DK mapping preserves the locality of fermionic interactions with an enlarged Hilbert space, thereby requiring additional constraints that define the physical sector. We incorporate these constraints directly into a Hamiltonian Variational Ansätz (HVA) through Clifford-gate state preparation and use the Variational Quantum Eigensolver (VQE) to show that the low-energy properties of the resulting qubit Hamiltonian are accurately reproduced. We further exploit particle-number conservation inherent in the ansätz to resolve distinct symmetry sectors and reliably access degenerate states. Consequently, we benchmark the DK-HVA against Jordan-Wigner-based variational circuits at nonzero chemical potential, where particle-hole symmetry and the associated half-filled sign-free condition are absent. Finally, we demonstrate the advantage of locality-preserving mappings in higher-dimensional fermionic systems, where the conventional Jordan-Wigner (JW) transformation generates increasingly long Pauli strings and corresponding circuit overheads. We further extend the framework to the spinful Fermi-Hubbard model and identify a tradeoff between fermionic-mode placement and the locality of hopping and on-site interaction terms. These results establish a practical framework combining locality-preserving fermion-to-qubit mappings, constraint-preserving Clifford state preparation, and symmetry-preserving variational circuits, that trades auxiliary-qubit and stabilizer-preparation overhead for reduced operator nonlocality, providing a practical route toward resource-efficient quantum simulation of higher-dimensional interacting fermionic systems.

quant-pharXiv:2609.16142•Sep 16, 2026•0 citations

Tree-Structured Commutativity Packing in Adaptive Variational Quantum Simulation: Measurement Overhead and Representation Limits

Hermawan Kresno Dipojono

While fermion-to-qubit isomorphisms are mathematically invariant under exact unitary statevector transformations, their physical implementation on noisy intermediate-scale quantum (NISQ) devices introduces severe, practical measurement bottlenecks. In this work, we systematically investigate the measurement efficiency of adaptive derivative-assembled pseudo-trotter variational quantum eigensolver (ADAPT-VQE) algorithms across three critical molecular benchmarks (LiH, BeH2, and H2O) at stretched bond lengths (2xRe) covering distinct point-group symmetries. First, we establish exact representation invariance across all target multi-reference geometries, demonstrating that operator pool gradients and statevector electronic energies remain strictly isomorphic under exact mathematical execution (E(LiH) = -7.691469 Ha, E(BeH2) = -15.110110 Ha, and E(H2O) = -74.566874 Ha). Second, we quantify the hardware measurement overhead by partitioning operator pool observables into qubit-wise commuting (QWC) tensor-product basis (TPB) cliques. We reveal that while both Jordan-Wigner (JW) and Bravyi-Kitaev (BK) encodings yield identical total Pauli term counts across the singlet excitation manifold (128 terms for H2O and BeH2), the hierarchical binary-tree layout of the BK mapping dramatically compresses the required measurement circuits---achieving up to a 45.31% reduction in total hardware execution steps compared to 12.50% under JW. Our findings demonstrate that tree-structured mappers serve as implicit, highly efficient measurement-packing engines for adaptive quantum simulations without altering the underlying variational trajectory.

quant-pharXiv:2606.13387•Jun 12, 2026•0 citations
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.
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_44​ in STO-3G with CAS(4e,4o), the mapped sector agrees with independent determinant FCI to 3.1×10−153.1\times10^{-15}3.1×10−15 Ha. At a nine-vector budget A-CASE has a 3.0193.0193.019 mHa error; replacing the determinant reference by a two-operator ADAPT-VQE state reduces it to 0.3420.3420.342 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 13.36413.36413.364 mHa at the nearest size match and 10.67410.67410.674 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.
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.
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 NNN spatial orbitals encoded via Jordan-Wigner, the XXX-rank, the binary rank of the XXX parts of its strings, of any commuting Pauli string family is no more than 2(N−1)2(N-1)2(N−1). Searches with the Robust Adaptive Nature-inspired Global Explorer (RANGE) reach this limit for CH4_44​, with the rank of the Hamiltonian support vouching for the maximality of the search values. And for all nonidentity Pauli strings on nnn 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+12^n+12n+1.
We investigate the utility of the locality-preserving Derby-Klassen (DK) fermion-to-qubit mapping [arXiv:2003.06939] for variational quantum simulation of two-dimensional ttt-VVV and Fermi-Hubbard models. The DK mapping preserves the locality of fermionic interactions with an enlarged Hilbert space, thereby requiring additional constraints that define the physical sector. We incorporate these constraints directly into a Hamiltonian Variational Ansätz (HVA) through Clifford-gate state preparation and use the Variational Quantum Eigensolver (VQE) to show that the low-energy properties of the resulting qubit Hamiltonian are accurately reproduced. We further exploit particle-number conservation inherent in the ansätz to resolve distinct symmetry sectors and reliably access degenerate states. Consequently, we benchmark the DK-HVA against Jordan-Wigner-based variational circuits at nonzero chemical potential, where particle-hole symmetry and the associated half-filled sign-free condition are absent. Finally, we demonstrate the advantage of locality-preserving mappings in higher-dimensional fermionic systems, where the conventional Jordan-Wigner (JW) transformation generates increasingly long Pauli strings and corresponding circuit overheads. We further extend the framework to the spinful Fermi-Hubbard model and identify a tradeoff between fermionic-mode placement and the locality of hopping and on-site interaction terms. These results establish a practical framework combining locality-preserving fermion-to-qubit mappings, constraint-preserving Clifford state preparation, and symmetry-preserving variational circuits, that trades auxiliary-qubit and stabilizer-preparation overhead for reduced operator nonlocality, providing a practical route toward resource-efficient quantum simulation of higher-dimensional interacting fermionic systems.
While fermion-to-qubit isomorphisms are mathematically invariant under exact unitary statevector transformations, their physical implementation on noisy intermediate-scale quantum (NISQ) devices introduces severe, practical measurement bottlenecks. In this work, we systematically investigate the measurement efficiency of adaptive derivative-assembled pseudo-trotter variational quantum eigensolver (ADAPT-VQE) algorithms across three critical molecular benchmarks (LiH, BeH2, and H2O) at stretched bond lengths (2xRe) covering distinct point-group symmetries. First, we establish exact representation invariance across all target multi-reference geometries, demonstrating that operator pool gradients and statevector electronic energies remain strictly isomorphic under exact mathematical execution (E(LiH) = -7.691469 Ha, E(BeH2) = -15.110110 Ha, and E(H2O) = -74.566874 Ha). Second, we quantify the hardware measurement overhead by partitioning operator pool observables into qubit-wise commuting (QWC) tensor-product basis (TPB) cliques. We reveal that while both Jordan-Wigner (JW) and Bravyi-Kitaev (BK) encodings yield identical total Pauli term counts across the singlet excitation manifold (128 terms for H2O and BeH2), the hierarchical binary-tree layout of the BK mapping dramatically compresses the required measurement circuits---achieving up to a 45.31% reduction in total hardware execution steps compared to 12.50% under JW. Our findings demonstrate that tree-structured mappers serve as implicit, highly efficient measurement-packing engines for adaptive quantum simulations without altering the underlying variational trajectory.