Extended Sample-Based Quantum Diagonalization

• Ext-SQD is a hybrid quantum–classical method that enriches a hardware‐sampled determinant subspace before classical Hamiltonian diagonalization.
• The approach leverages excitation operators, symmetry adaptation, amplitude amplification, and entanglement forging to enhance simulation accuracy.
• Benchmarks in quantum chemistry, lattice models, and embedding workflows demonstrate ext-SQD’s ability to deliver competitive accuracy with efficient resource usage.

Extended Sample-Based Quantum Diagonalization (ext-SQD) denotes a class of hybrid quantum–classical subspace-diagonalization methods built on Sample-Based Quantum Diagonalization (SQD). In SQD, a parameterized quantum state is sampled in the computational basis, the sampled bit-strings are mapped to Slater determinants, and the Hamiltonian is diagonalized classically in the sampled determinant subspace. ext-SQD retains this architecture but augments it by enlarging or restructuring the sampled subspace before a final classical diagonalization. In the cited literature, the label has been used for several non-identical extensions, including excitation-generated enlargements of sampled configuration spaces, symmetry-adapted closures under a space group, amplitude-amplified sampling protocols, and fragment-solver deployments in embedding workflows for quantum chemistry and materials modeling (Barison et al., 2024, Nogaki et al., 1 May 2025, Das et al., 29 Jun 2026, Stockinger et al., 4 May 2026).

1. Foundational formulation

The underlying object is a second-quantized electronic or lattice Hamiltonian expressed in an occupation-number basis. In electronic-structure applications, one representative form is

$$H = \sum_{pq\sigma} h_{pq}\,a_{p\sigma}^\dagger a_{q\sigma} + \frac12 \sum_{pqrs\sigma\tau} (pq|rs)\,a_{p\sigma}^\dagger a_{q\tau}^\dagger a_{s\tau} a_{r\sigma},$$

with fragment spin-orbitals mapped to qubits via Jordan–Wigner. Repeated measurements of a trial state in the computational basis yield bit-strings $x\in\{0,1\}^{2M}$, each encoding a Slater determinant $\lvert x\rangle$. Standard SQD forms a projector onto the sampled determinant set and diagonalizes the projected Hamiltonian. In one common notation, if $S_0=\{\lvert\psi_i\rangle\}$ is the list of distinct sampled determinants, then

$$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$

and approximate eigenpairs follow from a small generalized eigenproblem. When the determinant basis is orthonormal, the overlap matrix is the identity and the problem reduces to an ordinary eigenvalue problem (Nogaki et al., 1 May 2025, Das et al., 29 Jun 2026).

A central practical issue is that raw hardware samples often violate conserved quantum numbers. Several ext-SQD formulations therefore include a recovery stage. In the symmetry-adapted and molecular implementations, this may discard or probabilistically repair bit-strings to restore target particle numbers and spin sectors. The recovered configurations define the effective variational space, so the quality of ext-SQD depends jointly on the prepared trial state, the sampling statistics, and the structure imposed on the sampled determinant manifold (Barison et al., 2024, Nogaki et al., 1 May 2025).

2. Principal meanings of “ext-SQD”

Across the literature, the following extensions are explicitly described.

Variant	Defining extension	Representative papers
Excitation-augmented ext-SQD	Enlarge the sampled determinant space by acting with selected single and/or double excitation operators, then diagonalize the Hamiltonian in the enlarged computational-basis subspace	(Barison et al., 2024, Barroca et al., 13 Mar 2025, Shivpuje et al., 1 Oct 2025, Das et al., 29 Jun 2026)
Symmetry-adapted ext-SQD	Close the sampled subspace under space-group action or project sampled determinants into irreducible representations	(Nogaki et al., 1 May 2025)
SQD-AA	Use amplitude amplification to suppress already-seen bit-strings and make new ones more likely before each classical re-diagonalization	(Stockinger et al., 4 May 2026)
EF-integrated ext-SQD	Combine SQD with entanglement forging so that one qubit maps to a spatial orbital rather than a spin-orbital	(Smith et al., 11 Aug 2025)

The excitation-based formulation is the one most directly associated with the name “extended sample-based quantum diagonalization” in molecular applications. In that setting, a recovered configuration set $S=\{\lvert y_k\rangle\}$ is enlarged by acting with excitation operators $\hat E_I$ that map sampled determinants to new determinants,

$$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$

and the distinct $\lvert z_{Ik}\rangle$ are collected into an extended set $S_E$. The enlarged Hamiltonian $x\in\{0,1\}^{2M}$0 is then diagonalized by a standard eigenproblem, not the generalized QSE eigenproblem. A recurring point in this formulation is that ext-SQD never measures high-order reduced density matrices and adds no further quantum measurements; the extension is purely classical post-processing of hardware-sampled determinants (Barison et al., 2024).

A more specialized excitation-based realization was used for molten-salt fragments. There, after an iterative batch-and-recovery SQD loop converges, all determinants in the final lowest-energy batch with $x\in\{0,1\}^{2M}$1 are selected, all single excitations $x\in\{0,1\}^{2M}$2 are generated, and one final exact diagonalization in the enlarged subspace $x\in\{0,1\}^{2M}$3 yields the ext-SQD fragment energy and the one- and two-body RDMs for embedding reconstruction (Das et al., 29 Jun 2026).

The symmetry-adapted formulation modifies SQD in a different way. If the Hamiltonian commutes with a space group $x\in\{0,1\}^{2M}$4, a sampled subspace $x\in\{0,1\}^{2M}$5 need not be closed under $x\in\{0,1\}^{2M}$6. The method therefore constructs, for each sampled determinant, its orbit under the group and defines a symmetry-adapted space

$x\in\{0,1\}^{2M}$7

or, equivalently, uses explicit irrep projectors

$x\in\{0,1\}^{2M}$8

This extension is motivated by the observation that exact eigenstates lie in irreducible representations, whereas raw sampled spaces generally do not (Nogaki et al., 1 May 2025).

3. Trial states, recovery loops, and hardware realization

Most chemistry implementations use a Local Unitary Cluster Jastrow (LUCJ) ansatz. One one-layer form is

$x\in\{0,1\}^{2M}$9

where

$\lvert x\rangle$0

In the molten-salt study, the parameters were seeded from CCSD $\lvert x\rangle$1 and then classically optimized; in the excited-state study, LUCJ parameters were seeded from CCSD amplitudes without optimization. This variation reflects an implementation choice rather than a change in the subspace-diagonalization principle (Das et al., 29 Jun 2026, Barison et al., 2024).

The recovery loop is central to ext-SQD on noisy hardware. In the molten-salt deployment, each raw bit-string was repaired so that $\lvert x\rangle$2 and $\lvert x\rangle$3 by flipping minimal bits chosen in proportion to average occupations. The recovered pool was partitioned into $\lvert x\rangle$4 batches, and a subspace-diagonalization loop ran for $\lvert x\rangle$5 with $\lvert x\rangle$6. After each batch diagonalization, the global minimum energy was retained, orbital occupations were updated by averaging batch expectation values, and all determinants with squared CI coefficient at least $\lvert x\rangle$7 were carried into every batch of the next iteration. Convergence was declared when $\lvert x\rangle$8 Hartree and $\lvert x\rangle$9, or when $S_0=\{\lvert\psi_i\rangle\}$0 (Das et al., 29 Jun 2026).

A representative hardware implementation used IBM Heron r3 (“ibm_boston”) with heavy-hex connectivity and 130 qubits available. The qubit-to-spin-orbital map assigned two qubits per spatial orbital, arranged as two parallel zig-zag chains connected by ancilla qubits for $S_0=\{\lvert\psi_i\rangle\}$1–$S_0=\{\lvert\psi_i\rangle\}$2 Jastrow couplings. The largest fragment had $S_0=\{\lvert\psi_i\rangle\}$3 spatial orbitals, corresponding to 66 logical qubits; after transpilation, the circuit used approximately 1,400 two-qubit gates and depth approximately 600. Shot budgets were $S_0=\{\lvert\psi_i\rangle\}$4 for $S_0=\{\lvert\psi_i\rangle\}$5 and $S_0=\{\lvert\psi_i\rangle\}$6 for $S_0=\{\lvert\psi_i\rangle\}$7, with dynamical decoupling (XY4) and measurement twirling, but no zero-noise extrapolation or full purification (Das et al., 29 Jun 2026).

Related studies used analogous post-processing to mitigate hardware artifacts. In the multi-programming workflow for the LUCJ ansatz, the final ext-SQD energy difference relative to the HCI reference was reported as negligible, within $S_0=\{\lvert\psi_i\rangle\}$8 kcal/mol, despite deliberately randomized serial and parallel executions designed to probe cross-talk (Bazayeva et al., 12 May 2026).

4. Embedding, reduction, and application settings

A distinctive role of ext-SQD is as a solver inside larger embedding or reduction frameworks. In the embedded-wavefunction (EWF) workflow for FLiBe molten-salt clusters, conformations drawn from ab initio molecular dynamics were localized to IAOs, fragmented into atom-centered subspaces, and equipped with Schmidt baths constructed from RHF and augmented with MP2 natural orbitals. Fragments with $S_0=\{\lvert\psi_i\rangle\}$9 were solved by classical FCI, while larger fragments were solved by ext-SQD. Fragment energies and RDMs were then assembled through the partitioned-cumulant formula

$$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$0

Within this workflow, the classical and quantum EWF variants differed only in the fragment solver; embedding and reconstruction steps were otherwise identical across EWF-CCSD, EWF-FCI, and EWF-FCI+ext-SQD (Das et al., 29 Jun 2026).

Another embedding realization is DMET-SQD. There, a full-molecule RHF solution is localized, the environment block of the one-particle density matrix is diagonalized to define bath orbitals, and an impurity Hamiltonian is solved by SQD. In the reported demonstrations, 41- and 89-qubit full-molecule simulations were decomposed into 27- and 32-qubit active-region simulations on ibm_cleveland. This use of ext-SQD is not an enlargement by excitations; rather, it is a deployment of SQD-derived fragment solvers within density matrix embedding theory (Shajan et al., 2024).

Local embedding and active-space reduction also appear in battery surface chemistry. In the Li–O$$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$1 study, Density Difference Analysis was used to identify key orbitals, coupled-cluster natural orbitals refined the selection, and Ext-SQD enlarged the quantum-computed configuration set by single excitations. The same paper emphasized that quantum sampling cost and state preparation were inherited from SQD, whereas the extension was classical Q-SCI post-processing (Barroca et al., 13 Mar 2025).

Further integrations broaden the computational setting rather than the subspace rule itself. In entanglement-forged SQD, one qubit corresponds to a spatial orbital rather than a spin-orbital, reducing the required qubits by half. The EF wavefunction is written as

$$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$2

and sampled configurations are reassembled into a $$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$3-bit occupation basis before configuration recovery and subspace diagonalization. In implicit-solvent SQD, by contrast, the extension is entirely classical: the reaction-field operator from IEF-PCM modifies one-electron integrals while the quantum subspace-diagonalization machinery remains unchanged (Smith et al., 11 Aug 2025, Kaliakin et al., 14 Feb 2025).

5. Benchmark record

The most detailed chemistry benchmark for current hardware deployment is the molten-salt FLiBe study. Across nine 21-atom clusters, EWF-FCI+ext-SQD reproduced fragment relative energies with a mean absolute deviation of $$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$4 kcal/mol and a maximum error of $$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$5 kcal/mol relative to EWF-FCI. In absolute terms, ext-SQD fragment energies were approximately $$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$6–$$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$7 kcal/mol above FCI but nearly constant across conformations, so the offset canceled in relative energies. Tritium binding energies for 23-atom versus 22-atom clusters were reproduced to within $$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$8–$$P=\sum_{i=1}^{D}\lvert\psi_i\rangle\langle\psi_i\rvert, \qquad \tilde H = P H P,$$9 kcal/mol relative to embedded FCI/TCI references. The same study also found that fragmentation, not fragment solution, dominated the workflow error: conformational relative energies differed from full-system values by approximately $S=\{\lvert y_k\rangle\}$0 kcal/mol on average, up to approximately $S=\{\lvert y_k\rangle\}$1 kcal/mol, and tritium binding energies exhibited an approximately $S=\{\lvert y_k\rangle\}$2 kcal/mol fragmentation offset (Das et al., 29 Jun 2026).

Excited-state and strongly correlated benchmarks highlight a different strength of excitation-augmented ext-SQD. For N$S=\{\lvert y_k\rangle\}$3 in a $S=\{\lvert y_k\rangle\}$4 active space, SQD ground-state and $S=\{\lvert y_k\rangle\}$5 energies deviated by $S=\{\lvert y_k\rangle\}$6–$S=\{\lvert y_k\rangle\}$7 mHa from CASCI, QSE(SD) overestimated $S=\{\lvert y_k\rangle\}$8 by approximately $S=\{\lvert y_k\rangle\}$9–$\hat E_I$0 mHa, and ext-SQD with singles and doubles yielded $\hat E_I$1 and $\hat E_I$2 energies within $\hat E_I$3 mHa of CASCI across dissociation. In a $\hat E_I$4 active-space [2Fe-2S] cluster, SQD at $\hat E_I$5 had an error of approximately $\hat E_I$6 Ha, ext-SQD(SD) at $\hat E_I$7 reduced this to approximately $\hat E_I$8 Ha, and ext-SQD(SDT) reduced it further to approximately $\hat E_I$9 Ha; $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$0 and $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$1 lay within $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$2 mHa of HCI points of comparable $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$3 (Barison et al., 2024).

In materials and reaction benchmarks, the reported accuracy is likewise competitive within the chosen active spaces. For Li–O$$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$4 surface reaction calculations in a $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$5 space, Ext-SQD had product and reactant errors of $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$6 and $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$7 mHa, with average error $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$8 mHa, and the reaction energy was reported as $$\hat E_I\lvert y_k\rangle=\gamma_{Ik}\lvert z_{Ik}\rangle, \qquad \gamma_{Ik}\in\{0,\pm1\},$$9 eV compared with $\lvert z_{Ik}\rangle$0 eV. In diazirine and diazo photochemistry, SQD ground-state deviations were below $\lvert z_{Ik}\rangle$1 kcal/mol for all minima and transition states of parent diazirine relative to CASCI$\lvert z_{Ik}\rangle$2, while for phenyl-substituted diazirine in a $\lvert z_{Ik}\rangle$3 active space the SQD average deviation was approximately $\lvert z_{Ik}\rangle$4 kcal/mol relative to SCI; Ext-SQD reduced an excited-state error at the conical-intersection geometry from up to $\lvert z_{Ik}\rangle$5 kcal/mol to $\lvert z_{Ik}\rangle$6 kcal/mol (Barroca et al., 13 Mar 2025, Shivpuje et al., 1 Oct 2025).

Lattice-model and reduction-oriented benchmarks show additional, non-chemical performance regimes. In the two-leg ladder Hubbard model, momentum-basis ext-SQD with symmetry reached chemical accuracy of approximately $\lvert z_{Ik}\rangle$7 with subspace dimension $\lvert z_{Ik}\rangle$8, whereas the momentum basis without symmetry required about an order of magnitude larger $\lvert z_{Ik}\rangle$9; at $S_E$0, the reported dimensions were $S_E$1 for momentum-plus-symmetry, $S_E$2 for momentum without symmetry, $S_E$3 for the molecular basis without symmetry, and $S_E$4 for the molecular basis with symmetry. In hydrogen-abstraction simulations with entanglement forging, ext-SQD+EF gave total energies within $S_E$5 kcal mol$S_E$6 of DMRG for the $S_E$7 case and activation and reaction energies within $S_E$8 kcal of CCSD(T); in the larger $S_E$9 case, error cancellation reduced these energy-difference deviations to $x\in\{0,1\}^{2M}$00 kcal despite larger absolute total-energy errors. In implicit-solvent SQD/IEF-PCM, deviations from CASCI+IEF-PCM were reported as $x\in\{0,1\}^{2M}$01 kcal mol$x\in\{0,1\}^{2M}$02 for methanol, $x\in\{0,1\}^{2M}$03 for methylamine, $x\in\{0,1\}^{2M}$04 for ethanol, and $x\in\{0,1\}^{2M}$05 for water (Nogaki et al., 1 May 2025, Smith et al., 11 Aug 2025, Kaliakin et al., 14 Feb 2025).

Resource-oriented variants target sampling rather than post-processing accuracy. SQD-AA reported more than a factor-$x\in\{0,1\}^{2M}$06 reduction in total query complexity for algebraically and exponentially decaying model distributions; for an exponential tail with $x\in\{0,1\}^{2M}$07 and target $x\in\{0,1\}^{2M}$08, the reported values were approximately $x\in\{0,1\}^{2M}$09 queries for SQD-AA and approximately $x\in\{0,1\}^{2M}$10 for SQD, while for an algebraic tail with $x\in\{0,1\}^{2M}$11 the values were approximately $x\in\{0,1\}^{2M}$12 and approximately $x\in\{0,1\}^{2M}$13, respectively. The same study stated that, for all considered molecular examples, SQD-AA had the lowest total number of $x\in\{0,1\}^{2M}$14 gates while requiring circuits that were $x\in\{0,1\}^{2M}$15–$x\in\{0,1\}^{2M}$16 orders of magnitude shallower than those needed for iQPE (Stockinger et al., 4 May 2026).

6. Limitations, misconceptions, and open directions

A recurring misconception is that near-FCI fragment energies imply predictive full-workflow accuracy. The molten-salt study explicitly contradicts that interpretation: ext-SQD solved charged ionic fragments to near-FCI accuracy, but the dominant source of error in total energies arose from fragment construction. The paper therefore identified chemically informed bath thresholds, larger or adaptive fragments, inclusion of long-range dispersion, lower MP2 bath-truncation $x\in\{0,1\}^{2M}$17, and dynamic bath selection as essential directions for improvement. It also proposed integration into free-energy perturbation,

$x\in\{0,1\}^{2M}$18

to correct DFT free energies by quantum binding energies (Das et al., 29 Jun 2026).

Another limitation is that the term “ext-SQD” does not denote a single canonical algorithm. In excitation-augmented chemistry papers, the extension is a classical enlargement by singles, doubles, or higher excitations; in symmetry-adapted work, the extension is closure under group operations; in SQD-AA, the extension is a modified sampling loop using amplitude amplification. This suggests that the common invariant is not a specific operator set but the strategy of enriching the determinant subspace obtained from quantum sampling before classical diagonalization (Barison et al., 2024, Nogaki et al., 1 May 2025, Stockinger et al., 4 May 2026).

Classical post-processing is frequently the bottleneck. In one excited-state formulation, ext-SQD requires diagonalization of a $x\in\{0,1\}^{2M}$19 Hamiltonian with $x\in\{0,1\}^{2M}$20 when all singles and doubles are included; in the largest [2Fe-2S] case, $x\in\{0,1\}^{2M}$21 and $x\in\{0,1\}^{2M}$22. The battery study likewise identified classical memory and CPU time for handling $x\in\{0,1\}^{2M}$23 as the dominant bottleneck once the extension is applied. Several papers therefore propose pruning strategies, coefficient thresholds, automated excitation selection, or AI-driven orbital, fragment, and circuit optimization (Barison et al., 2024, Barroca et al., 13 Mar 2025, Das et al., 29 Jun 2026).

Method-specific caveats also remain. Symmetry adaptation depends strongly on basis choice: in the Hubbard benchmarks, symmetry improved convergence in the momentum basis but worsened it in the molecular-orbital basis because orbitals mixed poorly under translation. EF-integrated SQD currently enforces total spin only approximately and replaces the exact EF distribution by a compound approximation that can introduce bias. More broadly, future directions named in the cited papers include non-abelian point groups and SU(2), inclusion of triples and quadruples, transition dipoles and response functions, Green’s functions, lattice and solid-state applications, and hybrid post-processing with ph-AFQMC, which was reported to recover $x\in\{0,1\}^{2M}$24 mHa of correlation energy beyond SQD and to enable variance extrapolations to the $x\in\{0,1\}^{2M}$25 Ha level in favorable cases (Nogaki et al., 1 May 2025, Smith et al., 11 Aug 2025, Danilov et al., 7 Mar 2025).

In aggregate, ext-SQD is best understood as a quantum-centric determinant-subspace methodology whose defining operation is not variational energy minimization on the QPU but classical diagonalization in a subspace inferred, repaired, and then deliberately enriched from hardware samples. Its demonstrated roles range from low-lying excited states and lattice-model irreps to embedded fragment solvers for molten salts, radicals, solvated molecules, battery surfaces, and extended molecular systems (Barison et al., 2024, Das et al., 29 Jun 2026, Shajan et al., 2024).