Sample-Based Quantum Diagonalization (SQD)

• SQD is a quantum–classical hybrid framework that projects a many-body Hamiltonian onto a reduced, sampled subspace to estimate ground and excited state energies efficiently.
• It combines quantum hardware sampling with classical diagonalization and configuration recovery protocols to mitigate noise and reduce resource overhead.
• SQD has been applied in quantum chemistry and materials science, offering competitive accuracy and scalability for electronic structure and correlated lattice problems.

Sample-Based Quantum Diagonalization (SQD) is a quantum–classical hybrid algorithmic framework for estimating ground and excited state energies of many-body quantum systems, particularly relevant for electronic structure and correlated lattice problems. SQD leverages quantum hardware to sample many-body wavefunction configurations and then relies on classical processing—usually via subspace diagonalization—to extract physical observables within a reduced, physically motivated subspace. By replacing full Hilbert space diagonalization with the diagonalization of a carefully selected subspace, SQD significantly reduces the quantum and classical resource requirements required for accurate quantum simulation on near-term quantum hardware.

1. Algorithmic Structure and Projection Formalism

SQD operates by projecting the Hamiltonian $\hat{H}$, typically defined in a second-quantized form, onto a subspace $\mathcal{S} = \text{span} \{ |x_j\rangle \}$ constructed from a sampled set of configurations (usually Slater determinants encoded as computational basis states or bitstrings):

$$\hat{H}_\mathcal{S} = \mathcal{P}_\mathcal{S} \hat{H} \mathcal{P}_\mathcal{S}, \qquad \mathcal{P}_\mathcal{S} = \sum_{x \in \mathcal{S}} |x\rangle\langle x|$$

The electronic or lattice wavefunction is then expanded as:

$$|\psi\rangle = \sum_{x\in\mathcal{S}} \psi_x |x\rangle$$

Solving the projected eigenvalue equation

$\hat{H}_\mathcal{S} |\psi\rangle = E |\psi\rangle$

or, when non-orthogonality is present, the generalized form

$$Hc = ES c, \qquad S_{ij} = \langle x_i | x_j \rangle$$

returns low-energy eigenstates and associated energies with accuracy determined by the representability of the selected subspace $\mathcal{S}$.

Generation of the basis $\{ |x_j\rangle \}$ is accomplished via quantum circuit sampling of a trial state $|\Psi_\mathrm{tr}\rangle$, often derived from hardware-efficient ansätze such as the Local Unitary Cluster Jastrow (LUCJ) or other parametrizations informed by coupled-cluster amplitudes. Due to device noise or ansatz limitations, raw samples may violate required quantum numbers (particle number, spin projection). A self-consistent configuration recovery (S-CORE) protocol is applied iteratively: sampled bitstrings are filtered and repaired to match target quantum numbers, and occupation numbers $n_{p\sigma}$ are updated across Diagonalization–Sampling cycles to ensure consistency.

2. Subspace Construction, Sampling, and Configuration Recovery

Subspace construction in SQD is fundamentally stochastic. Multiple quantum measurement outcomes are gathered, forming sets $\tilde{\mathcal{X}}$ which are grouped into batches $\{\mathcal{S}^{(b)}\}$. Each batch yields a projected Hamiltonian, which is diagonalized classically (often by the Davidson algorithm). Configuration recovery proceeds by matching empirical occupation numbers with physical ones, typically employing a soft-constraint method for approximate symmetry restoration (e.g., through a penalty term $\lambda [S^2-s(s+1)]$ in open-shell problems).

The measurement bottleneck is governed by the quantum state's amplitude distribution over determinants. This is analytically mapped to a coupon-collector problem: estimating the expected number of measurement shots required to sample all important determinants (i.e., all bitstrings with $|c_I|^2 > \epsilon$ for some threshold) is a combinatorial problem controlled by the state's amplitude skew. For $m$ determinants, the expected number is

$$M = \sum_{r=0}^{m-1} (-1)^r \sum_{\substack{J\subseteq\{1,\dots,m\}\|J|=r}} \frac{1}{1 - \sum_{j\in J} p_j}$$

where $p_j$ is the probability for each determinant. In the uniform amplitude case, $M \sim m \log m$, but for distributions dominated by a few configurations (e.g., strongly single-reference), measurement overhead grows sharply (Reinholdt et al., 13 Jan 2025, Khamadja et al., 25 Sep 2025).

Redundancy in sampled configurations—i.e., repeatedly drawing already-seen determinants—introduces diminishing returns as the number of samples increases, limiting the efficiency of the method for high-accuracy, strongly correlated targets.

3. Extensions: Krylov, Randomization, and Symmetry Adaptation

Subspace design has significant impact on the convergence and feasibility of SQD. Beyond naïve sampling, the algorithm can be extended with:

• Sample-Based Krylov Quantum Diagonalization (SKQD): Instead of sampling only the trial state, one generates a Krylov subspace by applying short-time evolution:

$$|\psi_k\rangle = e^{-i k H t} |\psi_0\rangle,\, k = 0, \ldots, d-1$$

and samples each $|\psi_k\rangle$ to enrich the configuration set. SKQD comes with provable polynomial-time convergence so long as the ground state is "concentrated" (i.e., amplitude weight lies on $L \sim \mathrm{poly}(n)$ bitstrings with probabilities exceeding $\beta_L$) (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025).

• qDRIFT Randomized Compilation (SqDRIFT): To address the depth barrier of SKQD for complex Hamiltonians, the qDRIFT protocol constructs approximate time-evolution unitaries as random sequences:

$$V_{\mathbf{k}} = \prod_{j=1}^N \exp(-i h_{k_j} t\lambda/N)$$

with $k_j$ drawn according to Hamiltonian coefficients $c_i/\lambda$. This reduces expected circuit depth while controlling the approximation error $\epsilon \sim \mathcal{O}(\lambda^2 t^2/N)$. The union of outputs from many such randomized circuits forms the sampling pool for subspace diagonalization, preserving the convergence guarantees of SKQD under reasonable hardware constraints (Piccinelli et al., 4 Aug 2025).

• Symmetry-Adapted SQD: The subspace can be symmetrized to respect lattice, point-group, or particle-number symmetries by expanding each sampled configuration to its orbit under group actions:

$$\mathcal{S}_\mathrm{Sym} = \bigcup_{x\in\mathcal{S}} \mathcal{S}_x^{(g)}$$

with $\mathcal{S}_x^{(g)}$ obtained from group representation transformations $g|x\rangle$ and representation matrix $\mathcal{D}(g)$. Sparsity of $\mathcal{D}(g)$ (as in the momentum basis) ensures that the symmetrized subspace does not grow intractably and can dramatically improve energy convergence and the ability to capture key order parameters (e.g., superconducting correlations in ladder models) (Nogaki et al., 1 May 2025).

4. Applications and Performance in Quantum Chemistry and Materials Science

SQD has been applied across a range of molecular and materials problems:

• Molecules: SQD has been used to compute potential energy surfaces (PES) and excited states of dimers, open- and closed-shell molecules (e.g., CH$_2$), and challenging multi-reference systems. In each case, energies are recovered within a few milliHartree (often $\ll 1$ kcal/mol) of classical CASCI or HCI when the configuration pool is adequately sampled (Kaliakin et al., 11 Oct 2024, Barison et al., 1 Nov 2024, Liepuoniute et al., 7 Nov 2024).
• Materials: The method remains competitive for periodic systems and band gap computation by combining DFT-projected electronic structures with quantum circuit sampling, successfully demonstrating scalability up to nearly 50 qubits (Duriez et al., 13 Mar 2025).
• Embedded/Fragment Approaches: SQD has been incorporated with density-matrix embedding theory (DMET) to isolate active regions in large molecules, allowing efficient simulation of spatially extended systems—e.g., hydrogen atom rings and cyclohexane conformers—with resource counts compatible with current hardware (Shajan et al., 15 Nov 2024).
• Alchemical Free Energy Calculations: SQD-corrected configuration interaction is integrated within hybrid workflows for hydration free energy predictions, achieving FCI-quality corrections to MM-based paths (Bazayeva et al., 25 Jun 2025).

Benchmark results across these domains indicate that performance is dictated primarily by the completeness of the sampled subspace—which is ultimately limited by the stochastic sampling bottleneck discussed above. Configurations contributing small weight can dominate the correlation energy, necessitating advanced state-preparation or sampling techniques as system size and complexity increase.

5. Advantages, Limitations, and Hybrid Extensions

Advantages:

• Measurement Efficiency: SQD avoids the high cost of expectation value measurement over noncommuting Pauli operators required by VQE, focusing on stochastic projective measurements, with the quantum hardware acting purely as a sampling oracle.
• Robustness to Device Noise: Sampling is often more robust than variational optimization to hardware imperfections, and noise may inadvertently aid in discovering rare (important) configurations (Khamadja et al., 25 Sep 2025).
• Classical/Quantum Separation: All nontrivial diagonalization is performed classically in a reduced subspace, permitting classical certification, error mitigation (variance extrapolation), and further stochastic hybridization (e.g., SQD-generated trial functions for phaseless AFQMC (Danilov et al., 7 Mar 2025)).

Limitations:

• Sampling Inefficiency: The determinant discovery process (i.e., recovering all determinants with significant amplitude) is an instance of a coupon collector problem with nonuniform probabilities—highly skewed amplitude distributions dramatically increase measurement cost. There exists a trade-off between subspace size, compactness, and accuracy, quantified analytically (Reinholdt et al., 13 Jan 2025, Khamadja et al., 25 Sep 2025).
• Compactness vs. Discovery Rate Trade-off: Attempts to bias sampling toward less-discovered determinants (by adjusting probability distributions, e.g., flattening via $p_j^\alpha$) yield larger, less compact configuration expansions, increasing the classical post-processing cost. Keeping CI expansions compact slows new determinant discovery, impeding accuracy (Reinholdt et al., 13 Jan 2025).
• Scalability: For strongly correlated or large systems, neither hardware nor classical resources currently suffice to guarantee systematic convergence to chemical accuracy via pure SQD.

Hybridizations, such as using truncated SQD expansions as trial wave functions for phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) with variance extrapolation (Danilov et al., 7 Mar 2025), or combining with entanglement forging for qubit reduction (Smith et al., 11 Aug 2025), provide means to extend SQD's reach while mitigating core bottlenecks.

6. Future Directions and Outlook

The analytic identification of the sampling bottleneck and convergence guarantees via concentration and sparsity assumptions provide a rigorous roadmap for future development. Approaches for mitigating determinant redundancy and improving the coverage of rare but important configurations include:

• Constructing amplitude-flattening ansätze for state preparation.
• Applying importance sampling, active learning, or adaptive measurement allocation to focus on underrepresented determinants.
• Incorporating classical SCI heuristics in hybrid subspace expansion.
• Combining symmetry adaptation and qubit-reduction schemes to minimize hardware requirements (Khamadja et al., 25 Sep 2025).

A plausible implication is that as error-mitigation protocols, circuit compilation strategies (qDRIFT), and hybrid quantum–classical workflows mature, SQD and its extensions will establish the foundation for quantum advantage in correlated electron simulations—conditional on efficient sampling of all physically relevant configurations and the effective management of redundancy.

7. Comparative Table: SQD vs. Classical and Variational Quantum Approaches

Aspect	SQD/Core Approach	VQE/qEOM	Classical SCI
Quantum Role	Sampling (determinants)	Variational optimization	Deterministic selection
Measurement Cost	Bitstring sampling	Many Pauli bases	None
Bottleneck	Sampling redundancy, coverage	Optimization, measurement	Heuristic screening
Symmetry Restoration	Via configuration recovery/SCORE	Enforced by ansatz	Explicit criteria
Scalability	Limited by sampling, recovery	Limited by measurement/gradient	Set by heuristics, memory
Ground/Excited States	General, via extended subspace	Requires qEOM or expansion	Selected by design

SQD's main advantage—minimal measurement overhead—comes with the cost of scaling bottlenecks inherent to stochastic discovery and the classical diagonalization step. Its continued utility will depend on further algorithmic innovations addressing the interlocking challenges of sampling efficiency and subspace completeness.