Sample-Based Quantum Diagonalization (SQD)

• SQD is a hybrid quantum-classical method that approximates low-energy eigenstates by sampling key configurations from the Hilbert space.
• Its framework distinctly separates quantum sampling from classical Hamiltonian diagonalization, enhancing computational efficiency and resilience to noise.
• Variants such as Ext-SQD and half-qubit SQD extend the approach to excited states and resource reduction in electronic-structure and lattice models.

Sample-based Quantum Diagonalization (SQD), also referred to as Quantum Selected Configuration Interaction (QSCI), is a hybrid quantum-classical paradigm in which a quantum processor is used as a configuration sampler and a classical processor diagonalizes the Hamiltonian in the subspace spanned by the sampled configurations. In electronic-structure settings, the sampled computational-basis bitstrings encode Slater determinants, while in lattice settings they encode many-body occupation configurations. The central premise is that low-energy eigenstates can be approximated accurately from a reduced, physically relevant subset of the full Hilbert space, thereby replacing full-space diagonalization by subspace projection and classical selected-basis diagonalization (Reinholdt et al., 13 Jan 2025, McFarthing et al., 1 Feb 2026, Nogaki et al., 1 May 2025).

1. Definition and mathematical framework

SQD starts from the many-body eigenproblem

$\hat{H}|\Psi\rangle = E|\Psi\rangle,$

prepares an approximate state on quantum hardware, and samples computational-basis configurations $\mathbf{x}$ from the distribution

$$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$

From a sampled set $S$, the classical stage constructs the projector

$$\hat{P}_S=\sum_{\mathbf{x}\in S} |\mathbf{x}\rangle\langle \mathbf{x}|,$$

forms the restricted Hamiltonian

$\hat{H}_S=\hat{P}_S \hat{H}\hat{P}_S,$

and diagonalizes $\hat{H}_S$ to obtain approximate eigenpairs in the sampled subspace (McFarthing et al., 1 Feb 2026, Smith et al., 11 Aug 2025).

Equivalent formulations appear as the projected Schrödinger equation

$$\sum_n H_{mn} c_n = E c_m,\qquad H_{mn}=\langle x_m|\hat{H}|x_n\rangle,$$

with $|x_m\rangle$ drawn from quantum measurements. In chemistry, this turns SQD into a quantum-assisted selected-CI procedure: the quantum processor supplies determinants, and the classical processor performs the variational refinement. In strongly correlated model systems, the same formal structure applies to occupation-number basis states of lattice Hamiltonians (Smith et al., 11 Aug 2025, Park et al., 10 Mar 2026).

A defining feature of SQD is that the quantum and classical roles are sharply separated. The quantum device does not estimate the energy directly through repeated expectation-value measurements, but instead generates a determinant pool; the classically diagonalized subspace then supplies the variational energy and wavefunction coefficients. This separation underlies both the method’s practical robustness on noisy devices and its dependence on the representativeness of the sampled support (Kaliakin et al., 14 Feb 2025, Danilov et al., 7 Mar 2025).

2. State preparation, sampling, and recovery procedures

In much of the SQD literature, the sampled state is prepared with the Local Unitary Cluster Jastrow (LUCJ) ansatz. One commonly used form is

$$|\Psi \rangle = e^{-\hat{K}_2} e^{\hat{K}_1} e^{i\hat{J}_1} e^{-\hat{K}_1} |\Psi_{\mathrm{RHF}}\rangle,$$

where $\mathbf{x}$0 are one-body rotations, $\mathbf{x}$1 are density-density Jastrow operators, and $\mathbf{x}$2 is a Hartree-Fock reference. This choice reflects a recurring design principle in SQD: use a chemically motivated but hardware-manageable ansatz to bias measurements toward high-weight configurations (McFarthing et al., 1 Feb 2026, Kaliakin et al., 14 Feb 2025).

Raw samples obtained from NISQ hardware frequently violate conserved quantum numbers such as particle number or spin projection. The literature therefore introduces iterative recovery schemes—appearing under labels such as Self-consistent Configuration Recovery (SCCR) and S-CORE/S-CoRe—that repair or filter sampled bitstrings using occupation-number statistics extracted from prior subspace solutions. A representative update uses batchwise averages of one-body occupations,

$\mathbf{x}$3

and biases corrective bit flips toward the current occupancy profile (McFarthing et al., 1 Feb 2026, Kaliakin et al., 14 Feb 2025, Patra et al., 27 Nov 2025).

These recovery loops are not merely noise filters. In the extended implementations, they are integrated into an iterative workflow in which corrected samples define a subspace, diagonalization updates occupation vectors, and the revised occupations guide the next recovery step. In fragment and embedding settings, carryover mechanisms retain important configurations across iterations so that stochastic resampling does not erase previously discovered support (Wang et al., 16 Dec 2025, Park et al., 10 Mar 2026).

The same sampled-subspace idea has also been adapted to qubit-reduced formulations. In half-qubit SQD, only half of the spin-orbital sectors are simulated on the quantum device and the full determinant pool is assembled classically by tensoring sampled half-strings. Entanglement forging combined with SQD uses a related qubit-halving principle, mapping a qubit to a spatial orbital rather than a spin-orbital and reconstructing paired sectors classically (McFarthing et al., 1 Feb 2026, Smith et al., 11 Aug 2025).

3. Symmetry adaptation and basis dependence

A major methodological extension is symmetry-adapted SQD, which imposes space-group symmetry on the sampled subspace by enforcing

$\mathbf{x}$4

for the relevant symmetry operations $\mathbf{x}$5. If the Hamiltonian commutes with the symmetry group, its eigenstates transform according to irreducible representations, and a sampled subspace that does not respect this structure can degrade both accuracy and physical interpretability. Symmetry adaptation therefore enlarges or projects the sampled determinant pool so that the subspace is closed under the symmetry action (Nogaki et al., 1 May 2025).

The efficiency of this construction depends strongly on basis choice because the representation matrices $\mathbf{x}$6 of symmetry operations can be sparse or dense. For the two-leg ladder Hubbard model, the momentum basis automatically preserves translational symmetry in randomly sampled subspaces, while point-group operations act through highly sparse block structures; by contrast, in the molecular-orbital basis, translation acts through denser rotation blocks. The reported consequence is improved energy convergence in the momentum basis relative to the molecular-orbital basis for both the spin-quintet ground state and the spin-singlet excited state, together with rapid convergence of the superconducting correlation function and a demonstration of interaction-enhanced pairing correlations (Nogaki et al., 1 May 2025).

This line of work makes explicit a broader SQD principle: two distinct notions of compactness matter simultaneously. The first is wavefunction compactness, meaning concentration of the target state on relatively few determinants in the chosen basis. The second is symmetry-operator sparsity, meaning that symmetry completion of the sampled support does not proliferate configurations excessively. The paper’s formulation suggests that basis optimization for SQD is not only about making the state sparse, but also about making the symmetry algebra sparse (Nogaki et al., 1 May 2025).

4. Variant algorithms and subspace-engineering strategies

SQD has diversified into a family of related algorithms that modify how the determinant pool is generated, expanded, or filtered. Some variants primarily address excited states, some target resource reduction, and others attack the sampling bottleneck directly.

Variant	Defining mechanism	Representative reported effect
Symmetry-adapted SQD	Embeds translation and point-group symmetry into the sampled subspace	Improved energy convergence in the momentum basis; interaction-enhanced superconducting correlation reproduced (Nogaki et al., 1 May 2025)
Ext-SQD	Applies excitation operators to sampled configurations to expand the basis for low-lying states	Improves over original SQD and QSE(SD) for excited states of N$\mathbf{x}$7 and the [2Fe-2S] cluster (Barison et al., 2024)
HSQD / HCI-HSQD	Half-qubit sampling with classical reconstruction; optional HCI-inspired determinant selection	Matches SQD on N$\mathbf{x}$8 with half the qubits and 40% fewer measurements; yields sub-millihartree accuracy across the N$\mathbf{x}$9 potential energy surface (McFarthing et al., 1 Feb 2026)
CSQD	Clusters samples and uses cluster-specific reference occupancies for recovery	Lowers variational energies by up to 15.95 mHa for stretched N$$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$0 and up to 45.53 mHa for 2Fe-2S
SQD-AA / SqDRIFT	Uses amplitude amplification or qDRIFT-randomized Krylov propagation to improve sampling efficiency while preserving subspace logic	More than factor-100 query-complexity reduction for model distributions; convergence guarantees preserved in randomized Krylov form (Stockinger et al., 4 May 2026, Piccinelli et al., 4 Aug 2025)

Extended-SQD is the canonical excited-state generalization. Instead of diagonalizing only in the sampled determinant space, it augments the basis by acting on sampled configurations with single and double excitation operators, thereby creating a variationally freer expansion than QSE(SD) while retaining classically computable matrix elements. In benchmarks on N$$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$1, it was used to obtain the first singlet and triplet excited states; in the [2Fe-2S] cluster, it improved both energies and local observables relative to the original SQD basis (Barison et al., 2024).

Half-qubit SQD targets hardware-resource compression. In the reported formulation, the quantum circuit simulates only one spin sector, reducing the register from $$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$2 qubits to $$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$3 qubits. For N$$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$4 in a $$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$5 active space, the circuit resource comparison is 52 qubits and depth 905 for the full-qubit version versus 26 qubits and depth 429 for the half-qubit version; the same study reports 40% fewer measurements for comparable accuracy, subspaces 18–39% smaller than classical HCI in the HCI-HSQD variant, and active spaces up to $$p(\mathbf{x}) = |\langle \mathbf{x}|\Psi\rangle|^2.$$6 for iron-sulfur clusters (McFarthing et al., 1 Feb 2026).

Cluster-adaptive SQD modifies the recovery stage rather than the sampling stage. Standard SQD uses a single global reference occupancy vector; CSQD argues that this can be mixture-averaged in multimodal, strongly correlated regimes and therefore suppress mode-specific determinant structure. By clustering samples with unsupervised learning and performing recovery with cluster