---
title: Extended Sample-Based Quantum Diagonalization
url: https://www.emergentmind.com/topics/extended-sample-based-quantum-diagonalization-ext-sqd
type: topic
---

# Extended Sample-Based Quantum Diagonalization

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 [2411.00468][2505.00914][2606.30402][2605.02565].

## 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 [2505.00914][2606.30402].

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 [2411.00468][2505.00914].

## 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 | [2411.00468], [2503.10923], [2510.00484], [2606.30402] |
| Symmetry-adapted ext-SQD | Close the sampled subspace under space-group action or project sampled determinants into irreducible representations | [2505.00914] |
| SQD-AA | Use amplitude amplification to suppress already-seen bit-strings and make new ones more likely before each classical re-diagonalization | [2605.02565] |
| EF-integrated ext-SQD | Combine SQD with entanglement forging so that one qubit maps to a spatial orbital rather than a spin-orbital | [2508.08229] |

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 $H^E$ 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 [2411.00468].

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 $\lvert\langle x\vert\psi\rangle\rvert^2\ge 10^{-5}$ are selected, all single excitations $a_{p\sigma}^\dagger a_{q\sigma}\lvert x\rangle$ are generated, and one final exact diagonalization in the enlarged subspace $S_{\mathrm{ext}}$ yields the ext-SQD fragment energy and the one- and two-body RDMs for embedding reconstruction [2606.30402].

The symmetry-adapted formulation modifies SQD in a different way. If the Hamiltonian commutes with a space group $G$, a sampled subspace $S_0$ need not be closed under $G$. The method therefore constructs, for each sampled determinant, its orbit under the group and defines a symmetry-adapted space
$$
S_{\mathrm{sym}}=\bigcup_{x\in S_0} S_x^G,
$$
or, equivalently, uses explicit irrep projectors
$$
P^\alpha = \frac{d_\alpha}{|G|}\sum_g \chi^\alpha(g)^*\,g.
$$
This extension is motivated by the observation that exact eigenstates lie in irreducible representations, whereas raw sampled spaces generally do not [2505.00914].

## 3. Trial states, recovery loops, and hardware realization

Most chemistry implementations use a Local Unitary Cluster Jastrow (LUCJ) ansatz. One one-layer form is
$$
\lvert\Psi\rangle = e^{K(\theta)} e^{iJ(\theta)} e^{-K(\theta)} \lvert x_{\mathrm{RHF}}\rangle,
$$
where
$$
K=\sum_{pr\sigma}K_{pr}\,a_{p\sigma}^\dagger a_{r\sigma},
\qquad
J=\sum_{pr\sigma\tau}J_{p\sigma,r\tau}\,n_{p\sigma}n_{r\tau}.
$$
In the molten-salt study, the parameters were seeded from CCSD $T_1,T_2$ 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 [2606.30402][2411.00468].

The recovery loop is central to ext-SQD on noisy hardware. In the molten-salt deployment, each raw bit-string was repaired so that $N(x)=N_{\mathrm{ref}}$ and $S_z(x)=0$ by flipping minimal bits chosen in proportion to average occupations. The recovered pool was partitioned into $K=10$ batches, and a subspace-diagonalization loop ran for $t=1,\dots,T$ with $T=5$. 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 $10^{-4}$ were carried into every batch of the next iteration. Convergence was declared when $\lvert E^{(t)}-E^{(t-1)}\rvert<10^{-8}$ Hartree and $\max\lvert\Delta n_{p\sigma}\rvert<10^{-5}$, or when $t=T$ [2606.30402].

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 $\alpha$–$\beta$ Jastrow couplings. The largest fragment had $\mathrm{NORB}=33$ 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 $1\times10^5$ for $M<20$ and $1\times10^6$ for $M\ge 20$, with dynamical decoupling (XY4) and measurement twirling, but no zero-noise extrapolation or full purification [2606.30402].

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 $0.001$ kcal/mol, despite deliberately randomized serial and parallel executions designed to probe cross-talk [2605.12614].

## 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 $\mathrm{NORB}<13$ 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
$$
E_{\mathrm{total}}
=
E_{\mathrm{RHF}}
+
\mathrm{Tr}\!\left[F\sum_i \Delta\gamma_i^1\right]
+
\sum_i e_{2c}(\lambda_i^2).
$$
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 [2606.30402].

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 [2411.09861].

Local embedding and active-space reduction also appear in battery surface chemistry. In the Li–O$_2$ 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 [2503.10923].

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
$$
\lvert\Psi\rangle=\sum_\mu c_\mu\,\lvert u_\mu\rangle_\alpha\otimes\lvert v_\mu\rangle_\beta,
$$
and sampled configurations are reassembled into a $2M$-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 [2508.08229][2502.10189].

## 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 $0.30$ kcal/mol and a maximum error of $0.70$ kcal/mol relative to EWF-FCI. In absolute terms, ext-SQD fragment energies were approximately $2.1$–$2.9$ 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 $0.7$–$0.9$ 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 $12$ kcal/mol on average, up to approximately $30$ kcal/mol, and tritium binding energies exhibited an approximately $110$ kcal/mol fragmentation offset [2606.30402].

Excited-state and strongly correlated benchmarks highlight a different strength of excitation-augmented ext-SQD. For N$_2$ in a $(10e,8o)$ active space, SQD ground-state and $T_1$ energies deviated by $10$–$30$ mHa from CASCI, QSE(SD) overestimated $T_1$ by approximately $10$–$30$ mHa, and ext-SQD with singles and doubles yielded $T_1$ and $S_1$ energies within $1$ mHa of CASCI across dissociation. In a $(30e,20o)$ active-space [2Fe-2S] cluster, SQD at $D\approx1.67\times10^7$ had an error of approximately $+0.2$ Ha, ext-SQD(SD) at $D_E\approx1.26\times10^8$ reduced this to approximately $+0.075$ Ha, and ext-SQD(SDT) reduced it further to approximately $+0.05$ Ha; $S_1$ and $S_2$ lay within $<1$ mHa of HCI points of comparable $D_E$ [2411.00468].

In materials and reaction benchmarks, the reported accuracy is likewise competitive within the chosen active spaces. For Li–O$_2$ surface reaction calculations in a $(14e,20o)$ space, Ext-SQD had product and reactant errors of $0.8$ and $1.1$ mHa, with average error $1.0$ mHa, and the reaction energy was reported as $\Delta E_{\mathrm{Ext-SQD}}\approx -6.12$ eV compared with $\Delta E_{\mathrm{HCI}}\approx -6.30$ eV. In diazirine and diazo photochemistry, SQD ground-state deviations were below $1$ kcal/mol for all minima and transition states of parent diazirine relative to CASCI$(12,10)$, while for phenyl-substituted diazirine in a $(30,30)$ active space the SQD average deviation was approximately $1.1$ kcal/mol relative to SCI; Ext-SQD reduced an excited-state error at the conical-intersection geometry from up to $3$ kcal/mol to $\lesssim 2$ kcal/mol [2503.10923][2510.00484].

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 $10^{-3}t$ with subspace dimension $D\sim10^5$, whereas the momentum basis without symmetry required about an order of magnitude larger $D$; at $E_{\mathrm{error}}\approx10^{-3}t$, the reported dimensions were $4\times10^4$ for momentum-plus-symmetry, $3\times10^5$ for momentum without symmetry, $10^6$ for the molecular basis without symmetry, and $5\times10^6$ for the molecular basis with symmetry. In hydrogen-abstraction simulations with entanglement forging, ext-SQD+EF gave total energies within $1$ kcal mol$^{-1}$ of DMRG for the $(13e,13o)$ case and activation and reaction energies within $\lesssim1$ kcal of CCSD(T); in the larger $(23e,23o)$ case, error cancellation reduced these energy-difference deviations to $\lesssim2$ kcal despite larger absolute total-energy errors. In implicit-solvent SQD/IEF-PCM, deviations from CASCI+IEF-PCM were reported as $\le 0.06$ kcal mol$^{-1}$ for methanol, $0.05$ for methylamine, $0.35$ for ethanol, and $0.13$ for water [2505.00914][2508.08229][2502.10189].

Resource-oriented variants target sampling rather than post-processing accuracy. SQD-AA reported more than a factor-$100$ reduction in total query complexity for algebraically and exponentially decaying model distributions; for an exponential tail with $\alpha=1$ and target $m=50$, the reported values were approximately $5\times10^4$ queries for SQD-AA and approximately $10^7$ for SQD, while for an algebraic tail with $\gamma=5$ the values were approximately $3\times10^5$ and approximately $3\times10^7$, respectively. The same study stated that, for all considered molecular examples, SQD-AA had the lowest total number of $T$ gates while requiring circuits that were $3$–$4$ orders of magnitude shallower than those needed for iQPE [2605.02565].

## 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 $\eta$, and dynamic bath selection as essential directions for improvement. It also proposed integration into free-energy perturbation,
$$
\Delta\mu^{\mathrm{ex}} = -k_B T \ln \left\langle \exp[-\Delta\epsilon/k_B T]\right\rangle,
$$
to correct DFT free energies by quantum binding energies [2606.30402].

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 [2411.00468][2505.00914][2605.02565].

Classical post-processing is frequently the bottleneck. In one excited-state formulation, ext-SQD requires diagonalization of a $D_E\times D_E$ Hamiltonian with $D_E\lesssim D\cdot N_e\cdot(M-N_e)$ when all singles and doubles are included; in the largest [2Fe-2S] case, $D\approx1.67\times10^7$ and $D_E\approx1.26\times10^8$. The battery study likewise identified classical memory and CPU time for handling $D_E$ 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 [2411.00468][2503.10923][2606.30402].

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 $\mathcal O(100)$ mHa of correlation energy beyond SQD and to enable variance extrapolations to the $\mathcal O(10^{-3})$ Ha level in favorable cases [2505.00914][2508.08229][2503.05967].

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 [2411.00468][2606.30402][2411.09861].

Source: https://www.emergentmind.com/topics/extended-sample-based-quantum-diagonalization-ext-sqd