Sample-Based Krylov Quantum Diagonalization

• The paper introduces SBKQD, a hybrid algorithm that constructs a sparse Krylov subspace from projective measurements to efficiently estimate low-lying eigenvalues of many-body Hamiltonians.
• It combines time-evolved state sampling with classical Krylov methods to address scaling challenges of conventional quantum phase estimation and variational solvers.
• Innovations such as randomized compilation and mirror subspace diagonalization optimize measurement efficiency and mitigate errors on near-term quantum devices.

Sample-based Krylov quantum diagonalization (SBKQD, also referred to as SKQD) is a class of hybrid quantum–classical algorithms for eigenvalue estimation of large quantum many-body Hamiltonians. SBKQD leverages quantum resources to generate a sparse, physically meaningful subspace—built from bitstrings sampled from time-evolved states—that captures essential properties of the low-lying spectrum. Classical diagonalization is then performed in this reduced subspace, combining the long-standing power of Krylov/lanczos methods with sample efficiency suitable for near-term and early fault-tolerant quantum devices. SBKQD and its variants address the scaling bottlenecks of conventional quantum phase estimation and variational eigenvalue solvers by enabling polynomial-time convergence given ground-state sparsity and carefully designed sampling and measurement protocols.

1. Mathematical Framework of Sample-Based Krylov Diagonalization

Let $H$ be an $n$-qubit Hamiltonian, $|ψ_0⟩$ a reference state with sufficient overlap on the low-energy manifold, and $U = e^{-i H \Delta t}$ a fixed timestep propagator. The $D$-dimensional unitary Krylov subspace is

$$\mathcal{K}_D(H,|ψ_0⟩) = \operatorname{Span}\{|ψ_j⟩ = U^j|ψ_0⟩\},\quad\text{for } j=0,\ldots,D-1$$

Classically, one forms the overlap and projected Hamiltonian matrices

$S_{ij} = ⟨ψ_i|ψ_j⟩,\qquad H_{ij} = ⟨ψ_i|H|ψ_j⟩,$

and solves the generalized eigenvalue problem $Hc = E S c$ for the smallest eigenvalue $E$, approximating the ground-state energy.

In SBKQD, rather than measuring all elements $S_{ij},\ H_{ij}$ via deep Hadamard-test circuits—which require controlled unitaries and an ancilla—one samples projectively from each Krylov vector $|ψ_j⟩$, collecting bitstrings in the computational basis, and forms the projected Hamiltonian by restricting $H$ to the set of unique sampled determinants. The classical diagonalization is then performed in this empirical subspace, which is typically much smaller than the full Hilbert space but still capable of capturing the essential spectral information, provided the ground state is sparse or "concentrated" (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025, Firt et al., 19 Dec 2025, Lee et al., 22 Dec 2025).

2. Quantum–Classical Hybrid Workflows

A typical SBKQD protocol consists of the following steps:

1. Reference State Preparation: Initialize $|ψ_0⟩$ with non-vanishing ground-state overlap, e.g., Hartree–Fock for molecules or product/sector states for spin models.
2. Krylov Step Preparation: For $j=0,\ldots,D-1$, use time evolution or block-encoding to prepare $|ψ_j⟩ = U^j|ψ_0⟩$.
3. Projective Measurement: For each $|ψ_j⟩$, measure the register $M$ times in the computational basis, collecting bitstrings $\{x_j^{(m)}\}$.
4. Sampled Subspace Assembly: The union of unique bitstrings across all $j$ forms the empirical subspace $\mathcal{B}$.
5. Hamiltonian Projection and Diagonalization: Compute (classically) the restriction $H_{\mathcal{B}}$ on $\operatorname{Span}\{|b⟩ : b \in \mathcal{B}\}$ and solve the eigenvalue problem for its lowest root (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025).

This approach is robust to hardware noise—since only bitstring frequencies are used—and avoids the need for controlled evolution. Extensions like Partitioned Quantum Subspace Expansion (PQSE) further decompose the process into variance-optimized iterative steps, improving stability and reducing circuit depth (O'Leary et al., 2024).

3. Measurement-Efficient and Hardware-Tailored Protocols

Standard SBKQD has significant sample complexity when naively implemented. Several innovations address measurement efficiency:

• Real-Time Approaches and Toeplitz Structure: Utilizing real-time evolution $|ψ_j⟩ = U^j|ψ_0⟩$ and the resulting Toeplitz/Hankel structure of the projected matrices reduces the number of independent matrix elements to $O(D)$ (Yoshioka et al., 2024, Stair et al., 2022, Zhang et al., 2023).
• Krylov Time Reversal (KTR): For time-reversal symmetric Hamiltonians $H$ with a stabilizer $T$, B and A matrices can be obtained from real expectation values of $T$ and $iHT$ on the state at half the time between basis elements, eliminating controlled unitaries/ancilla, and reducing circuit depth (Mariella et al., 30 Jul 2025).
• Randomized Compilation (SqDRIFT): Replaces deep, deterministic Trotterization with qDRIFT-style stochastic sequences, preserving rigorous convergence guarantees while making quantum chemistry Hamiltonians tractable on current hardware (Piccinelli et al., 4 Aug 2025, Stair et al., 2022).
• Mirror Subspace Diagonalization (MSD): Expresses $H$ as a linear combination of time-shifted unitaries via central finite-difference formulas, enabling optimal allocation of sampling resources and achieving near-theoretical minimum measurement cost for energy estimation in strongly correlated molecules (Kanasugi et al., 26 Nov 2025).
• Shifting and Coefficient Splitting: Systematic elimination of redundant Hamiltonian contributions and optimal variance allocation across measurement fragments leads to 20–500$\times$ fewer needed measurements for subspace matrix element estimation in chemical settings (Lee et al., 2024).

These techniques, when combined with symmetry exploitation and error-mitigation strategies (e.g., post-selection on conserved quantities), enable SBKQD to function at scale—even on moderate-depth NISQ or early-fault-tolerant quantum devices.

4. Error Analysis, Convergence Guarantees, and Conditioning

The convergence of SBKQD to the true ground-state energy is governed by properties of the subspace and sampling:

• Sparsity-Driven Success: Provided the true ground state $|GS⟩$ is $(\alpha_L,\beta_L)$-concentrated on $L$ computational basis states and that $|ψ_0⟩$ has overlap $|\gamma_0|$ with $|GS⟩$, sample complexity scales as $O(D^2/(|\gamma_0|^2 \beta_L)\log(L/\eta))$ for success probability $1-\eta$. The subspace diagonalization error decays as $\sqrt{8} \|H\| (1-\sqrt{\alpha_L})^{1/2}$ (Yu et al., 16 Jan 2025, Firt et al., 19 Dec 2025, Piccinelli et al., 4 Aug 2025).
• Finite Sampling Noise: The projected matrices inherit statistical fluctuations from measurement. Mathematically, each entry's error is modeled as (complex) Gaussian, and total spectral error is analyzed via matrix concentration and generalized eigenproblem perturbation bounds (Lee et al., 2023). The key practical protocol is to discard ill-conditioned directions in the overlap matrix $S$ (thresholding small singular values at the scale of statistical error), ensuring eigenvalue stability (Lee et al., 2023, O'Leary et al., 2024).
• Resource and Circuit Scaling: For fixed precision $\epsilon$ in each matrix entry and subspace dimension $D$, total shot cost is $O(D^2/\epsilon^2)$ (naive), but lower for optimized schemes: $O(D \log D)$ for Toeplitz-based protocols or $O(D^2 \log D / \eta^2)$ for central-difference-based schemes (MSD). For quantum chemistry, classical diagonalization in the sampled subspace is sub-exponential in system size if ground-state sparsity is maintained, but bottlenecks arise as subspace grows.

The choice of Krylov subspace functions—e.g., powers of $H$, Chebyshev polynomials, or filtered/gaussianized variants—can exponentially improve subspace conditioning, reducing the exponential scaling penalty in large systems (Zhang et al., 2023).

5. Algorithmic Variants and Their Properties

Multiple variants of SBKQD have emerged, each addressing specific implementation or computational challenges:

Variant / Technique	Key Features	Best Use Cases
Krylov Time Reversal (KTR) (Mariella et al., 30 Jul 2025)	Leverages time-reversal symmetry, uses only single-observable measurements, no ancilla/control	Spin and lattice models with anti commuting symmetry
SqDRIFT (Piccinelli et al., 4 Aug 2025)	Randomized LCU compilation for Trotterization	Deep circuits, electronic structure, fault-tolerant NISQ
Mirror Subspace Diagonalization (MSD) (Kanasugi et al., 26 Nov 2025)	Finite-difference estimation, optimal sample cost	Strongly correlated molecules, large basis sets
Generative KSR (GenKSR) (Lee et al., 22 Dec 2025)	Trains a classical generative model on measurement outcomes, eliminates need for repeated QPU calls	High-throughput quantum chemistry; many Hamiltonians
Partitioned QSE (PQSE) (O'Leary et al., 2024)	Sequential, variance-driven subspace expansion	Improves robustness to sampling noise
Gaussian-power basis (Zhang et al., 2023)	Integrates ground-state filtering for superior conditioning	Large $D$, noise sensitive applications

Each variant enables different trade-offs between circuit depth, number of quantum measurements (shots), and classical postprocessing, and may employ specific hardware-aware strategies, e.g., all-to-all connectivity, error-mitigation post-selection, or stochastic compilers.

6. Practical Applications and Quantum Hardware Results

SBKQD and its variants have demonstrated practical utility in several domains:

• Gauge Theories: SKQD accurately captures ground-state energy structure and order parameters in lattice Schwinger models, reducing Hilbert space by up to 80%, and reliably reproducing phase transitions on trapped-ion and superconducting processors (Rosanowski et al., 30 Oct 2025).
• Molecular Electronic Structure: SQD and SqDRIFT compute ground-state energies in active spaces as large as 48 qubits (e.g., coronene), achieving errors competitive with CISD/CCSD references, and only requiring moderate-depth circuits (Piccinelli et al., 4 Aug 2025, Wray et al., 4 Dec 2025).
• Strongly Correlated and Spin Systems: SKQD has been benchmarked on 30-qubit Heisenberg chains, capturing magnetization curves consistent with DMRG, and demonstrating scaling to an 85-qubit single-impurity Anderson model—beyond classical exact diagonalization limits (Firt et al., 19 Dec 2025, Yu et al., 16 Jan 2025).
• Noise and Error Mitigation: Light error-mitigation strategies (e.g., bitstring validity post-selection, enforcement of sector constraints) enhance empirical convergence beyond noise-free simulation in some cases (Wray et al., 4 Dec 2025).

Measured quantum resource requirements (circuit depth, shot count, classical memory) and observed accuracy depend sensitively on ground-state sparsity, subspace conditioning, and the degree to which advanced measurement allocation (e.g., shifting, coefficient splitting) is utilized.

7. Limitations, Open Challenges, and Future Perspectives

Despite demonstrated advances, SBKQD faces inherent and practical limitations:

• Scaling with Dense Ground States: In regimes where the ground state is highly delocalized (low participation ratio), the required number of samples and necessary subspace size grow rapidly, rendering SBKQD less efficient (Firt et al., 19 Dec 2025).
• Classical Postprocessing Bottleneck: Diagonalization in the sampled subspace becomes expensive as the number of unique bitstrings grows exponentially, although this is partially mitigated by low-weight initializations and advanced sampling (Rosanowski et al., 30 Oct 2025).
• Measurement Overhead: Even after sample-efficient protocols, the $1/\sqrt{M}$ convergence of sampling error versus the exponential sensitivity of the generalized eigenproblem can dominate for large subspaces and low error targets (Lee et al., 2024, Lee et al., 2023).
• Ill-conditioning and Numerical Stability: Overlap matrices can become near-singular at large Krylov order or inappropriate step size. Regularization and basis-thresholding techniques are essential for practical stability (Lee et al., 2023, O'Leary et al., 2024).
• Extensibility to Excited State and Dynamics: While straightforward (through block-Krylov or basis extension to excited-state references), further work is necessary for robust integration in SBKQD frameworks (Oumarou et al., 9 Jan 2025, Lee et al., 22 Dec 2025).

Future directions include:

• Generative models (GenKSR) to fully classicalize the sampling step (Lee et al., 22 Dec 2025)
• Enhanced error-mitigation, adaptive sampling, and block-Krylov subspace recycling
• Application to 2+1D or non-Abelian gauge theories (Rosanowski et al., 30 Oct 2025)
• Empirical and theoretical studies of the asymptotics in dense or "area-law" ground states

Sample-based Krylov quantum diagonalization thus constitutes a robust, flexible platform for quantum eigenvalue estimation with strong theoretical foundations, a suite of algorithmic accelerations, and a growing record of practical deployment across condensed matter, quantum chemistry, and gauge theory models (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025, Kanasugi et al., 26 Nov 2025, Piccinelli et al., 4 Aug 2025, Mariella et al., 30 Jul 2025, Firt et al., 19 Dec 2025, Lee et al., 2023).