Sample-Based Quantum Diagonalization

• Sample-based quantum diagonalization methods are hybrid algorithms that project complex quantum problems into a manageable subspace using quantum sampling.
• They employ techniques like Krylov subspaces, randomized evolution, and neural network-driven basis adaptation to achieve near chemical accuracy with reduced quantum resources.
• These methods balance classical diagonalization with efficient quantum sampling to address spectral extraction challenges in quantum chemistry, materials, and condensed matter physics.

Sample-based quantum diagonalization methods constitute a class of hybrid quantum–classical algorithms for extracting the spectral features—eigenvalues and eigenvectors—of quantum operators and states from limited quantum resources. These methods circumvent the exponential complexity of direct full-space diagonalization by employing quantum devices as sampling engines, generating a strategically selected subspace in which the Hamiltonian or target density operator is then diagonalized using classical computation. Variants within this paradigm include techniques that leverage quantum circuit sampling (SQD), Krylov subspaces, adaptive neural basis generation, symmetry adaptation, and randomized basis evolution, targeting both ground and excited state problems across chemistry, materials science, and condensed matter physics.

1. Core Principles of Sample-Based Quantum Diagonalization

At the foundation, sample-based quantum diagonalization replaces the aim of direct quantum simulation of entire Hilbert spaces with a projection of the problem into a physically relevant subspace. This subspace is defined either by probabilistic sampling from a quantum-prepared state (Khamadja et al., 25 Sep 2025), time-propagated Krylov vectors (Yu et al., 16 Jan 2025), neural-network-generated configurations (Cantori et al., 18 Aug 2025), or symmetry-adapted bitstrings (Nogaki et al., 1 May 2025).

A canonical SQD workflow consists of the following steps:

1. State Preparation: A quantum circuit prepares a variational state, often using ansatzes such as local unitary cluster Jastrow (LUCJ) or Unitary Coupled Cluster (UCC) types.
2. Sampling: The quantum device is measured in the computational basis, producing bitstrings/configurations with probabilities $\mathbb{P}_\Psi(x) = |\langle x | \Psi \rangle|^2$.
3. Subspace Projection: The collected configurations $\chi = \{x_1, x_2, \ldots, x_d\}$ form a subspace $\mathcal{S}$; the Hamiltonian is projected as $H_\mathcal{S} = P_\mathcal{S} H P_\mathcal{S}$ with $$P_\mathcal{S} = \sum_{x\in\mathcal{S}} |x\rangle\langle x|$$.
4. Classical Diagonalization: The projected matrix is diagonalized (e.g., Davidson's method), yielding approximate eigenpairs.

This approach relies on the concentration of the target quantum state: namely, that relevant eigenstates (ground or excited) have most of their weight on a polynomially sized set of configurations within the full, exponentially large Hilbert space (Piccinelli et al., 4 Aug 2025). Such concentration allows for efficient sampling, with rigorous error bounds and probabilistic guarantees under appropriate assumptions.

2. Key Variants and Algorithmic Extensions

Numerous variants and enhancements of SQD have emerged:

• Sample-based Krylov Quantum Diagonalization (SKQD): This method uses time-evolved Krylov vectors $|\psi_k\rangle = e^{-ikH\Delta t}|\psi_0\rangle$ as sampling sources, proven to yield polynomial convergence if the ground state is concentrated (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025).
• Randomized Evolution (SqDRIFT): Combining SKQD with the qDRIFT randomized compilation, this reduces circuit-depth requirements by simulating effective Hamiltonian evolution as a stochastic sequence of elementary terms, maintaining convergence guarantees while increasing the feasibility for noisy, shallow hardware (Piccinelli et al., 4 Aug 2025).
• Extended SQD (Ext-SQD): Augments the sampled subspace by applying excitation operators (singles, doubles, triples) to configurations already present, allowing efficient computation of both ground and excited states, even in systems with strong multireference character (Barison et al., 1 Nov 2024, Shivpuje et al., 1 Oct 2025, Barroca et al., 13 Mar 2025).
• Symmetry-Adapted SQD: The bitstring subspace is postprocessed to ensure closure under the relevant symmetry group, e.g., lattice translations or point-group operations, which enhances sampling efficiency and energy convergence, especially in momentum bases (Nogaki et al., 1 May 2025).
• Neural-Network-Driven SBD (AB-SND): Autoregressive neural networks generate an efficient proposal distribution and, with an additionally optimized basis rotation $U(\theta)$, mitigate the exponential scaling typical in delocalized or non-concentrated ground states (Cantori et al., 18 Aug 2025).
• Error Reduction in Krylov Subspace Methods: For matrix elements estimated from quantum measurements, techniques such as shift preconditioning and iterative coefficient splitting (ICS) have been developed to minimize the measurement budget and suppress error propagation in generalized eigenvalue problems (Lee et al., 4 Sep 2024).

3. Analytical and Practical Bottlenecks

A distinguishing feature of sample-based methods is the explicit description of their measurement bottleneck. Discovery of the full relevant set of Slater determinants is mathematically equivalent to a non-uniform coupon-collector problem. The expected number of projective measurements $M$ required to observe all $m$ important determinants with probability vector $p = (p_1,\ldots,p_m)$ is

$$M = \sum_{r=0}^{m-1} (-1)^r \sum_{J\subset S, |J|=r} \frac{1}{1-\sum_{i\in J} p_i}$$

where the denominator is dominated by rare determinants (small $p_i$), quantifying the resource overhead due to highly skewed distributions (Khamadja et al., 25 Sep 2025). For uniform $p_i$ one recovers $M\sim m(\ln m + \gamma)$ with $H_m$ the harmonic number and $\gamma$ Euler’s constant.

Moreover, substantial numerical evidence shows that SQD can reproduce full configuration interaction (FCI) or CASCI energies to within chemical accuracy using configuration subspaces that are several orders of magnitude smaller than the full space—for example, achieving $<1$ kcal/mol average error for (30e,30o) problems relative to SCI benchmarks (Shivpuje et al., 1 Oct 2025). However, if determinant amplitudes are sharply peaked (e.g., dominated by the Hartree–Fock state), the discovery efficiency drops rapidly with sample number, presenting a severe scaling bottleneck that has been validated both in simulation and on hardware (Reinholdt et al., 13 Jan 2025, Khamadja et al., 25 Sep 2025).

4. Applications and Demonstrated Results

Sample-based quantum diagonalization has been validated across a range of quantum simulation challenges:

• Quantum Chemistry: Calculation of ground and excited-state energies, potential energy surfaces, and conical intersections for diazirine/diazo photochemistry using (12,10) to (30,30) active spaces, reproducing CASCI and SCI trends (Shivpuje et al., 1 Oct 2025). Combined with error mitigation (e.g., S-CORE, hardware twirling), chemical accuracy is achieved even on superconducting processors for systems up to 77 qubits (Barison et al., 1 Nov 2024).
• Periodic Materials: Computation of band gaps in crystalline silicon and zirconia with extended Hubbard models, demonstrating scalability to over 40 qubits and energy deviations within 50 meV compared to HCI (Duriez et al., 13 Mar 2025).
• Surface Reactions: Ext-SQD enables ground-state energy prediction in surface-adsorbed battery reactions (oxygen reduction on Li electrodes), yielding reaction energies that match CASCI and surpass CCSD (Barroca et al., 13 Mar 2025).
• Quantum Many-Body Lattice Models: Symmetry-adapted SQD applied to ladder Hubbard models captures both ground and excited state energies and accurately reproduces superconducting correlation decay, aided by compact subspace construction in momentum space (Nogaki et al., 1 May 2025).
• Quantum Dynamics and Spectroscopy: Variants such as VQSD offer a route to extract entanglement spectra in Heisenberg chains and enable applications in entanglement spectroscopy and quantum principal component analysis (LaRose et al., 2018).
• Hybrid Embedding: The DMET-SQD integration allows treatment of extended molecules (e.g., 89-qubit cyclohexane) by reducing the quantum subproblem to active fragments and their baths, with energy differences agreeing within 1 kcal/mol compared to classical DMET-FCI (Shajan et al., 15 Nov 2024).
• Chemical Reactions with Implicit Solvent: SQD combined with IEF-PCM models produces solvation energy corrections within 0.1 kcal/mol of CASCI/PCM for prototypical molecules with up to 52 qubits, indicating applicability in biochemical environments (Kaliakin et al., 14 Feb 2025).
• Combined Quantum–Classical Post-Processing: Use of phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) with SQD trial wavefunctions recovers up to 100 mHa of correlation energy missed by SQD alone, with extrapolation to the zero-variance limit further improving accuracy (Danilov et al., 7 Mar 2025).

5. Limitations, Trade-Offs, and Scaling Considerations

While SQD and its descendants bypass certain quantum resource bottlenecks (notably, circuit depth and variational barren plateaus in VQE), they introduce rigorous limitations:

• Sampling Bottleneck: The transition from identifying the dominant amplitudes to achieving full ground-state support can lead to an exponential increase in measurement cost, particularly for states with highly non-uniform amplitudes. Attempts to alter the sampling distribution (e.g., power-law scaling) can reduce discovery time but result in noncompact, less accurate CI expansions (Reinholdt et al., 13 Jan 2025).
• Subspace Compactness vs. Accuracy: Compared to classical selected CI such as HCI, SQD-generated expansions are less compact, leading to more expensive subsequent classical diagonalization (Reinholdt et al., 13 Jan 2025). This tension is central in determining the utility of SQD for large, chemically relevant systems.
• Noise and Recovery: Finite sampling and quantum noise can result in subspaces populated by erroneous or symmetry-violating configurations; elaborate self-consistent recovery and projection procedures (S-CORE, symmetry adaptation) are required to restore physical constraints, at the expense of additional postprocessing (Shivpuje et al., 1 Oct 2025, Nogaki et al., 1 May 2025).
• Hardware Constraints: Certain variants (SKQD, SqDRIFT) are bounded by the depth of time-evolution circuits achievable on current hardware. Use of randomized evolution protocols such as qDRIFT can alleviate these constraints, allowing for utility-scale runs on complex chemical Hamiltonians with preserved convergence guarantees (Piccinelli et al., 4 Aug 2025).

6. Connections to Machine Learning and Adaptive Sampling

Recent approaches directly employ machine learning strategies to enhance the configuration selection and basis concentration:

• Neural Network Sampling: Autoregressive neural networks (SND, AB-SND) generate configuration batches that maximize coverage of the important support in the ground state, effectively adapting the sampling strategy to the underlying quantum system (Cantori et al., 18 Aug 2025).
• Adaptive Basis Optimization: By parameterizing and optimizing the basis transformation $U(\theta)$, one can rotate the computational basis to concentrate the ground-state weight, crucial in delocalized or critical regimes where standard sampling is inefficient (Cantori et al., 18 Aug 2025).
• Hybrid Strategies: The sample-based diagonalization paradigm is compatible with future hybrid quantum–classical workflows, including the explicit implementation of global unitaries on hardware.

7. Outlook and Ongoing Challenges

Sample-based quantum diagonalization methods represent a convergence of quantum computing, numerical linear algebra, and computational science, providing scalable, rigorous alternatives to energy estimation in the NISQ and early fault-tolerant eras. Advances in circuit compilation (randomized evolution), basis adaptation (neural rotation or symmetry embedding), statistical postprocessing, and hardware error mitigation have collectively made possible chemically accurate quantum simulations of molecules, periodic materials, and correlated lattice models at scales previously inaccessible.

Fundamental bottlenecks—manifested in sampling efficiency, configuration compactness, determinant recovery, and measurement cost—are now characterized both analytically and numerically. While the methods succeed in regimes of state concentration and moderate Hilbert space size, further improvements in quantum state preparation, sampling efficiency (possibly via active learning or adaptive proposals), and integration with stochastic classical solvers (e.g., ph-AFQMC) will be required to fully realize their potential for large-scale, strongly correlated quantum systems. The mapping of the configuration discovery process to the coupon collector problem provides a rigorous roadmap for resource forecasting and motivates continued innovation in both quantum and classical components of these algorithms.

These developments reinforce the view that sample-based quantum diagonalization, and its evolving algorithmic variants, will constitute a central pillar of quantum computational practice in quantum chemistry, condensed matter, and materials theory throughout the utility regime of quantum hardware.