SKQD: Sample-Based Krylov Quantum Diagonalization

• SKQD is a hybrid quantum-classical algorithm that constructs a Krylov subspace via short-time evolutions and computational-basis sampling to approximate ground-state energies.
• It replaces expensive matrix-element estimation with efficient measurement of time-evolved states and subsequent classical diagonalization, reducing circuit depth.
• The method offers noise robustness and scalability under specific sparsity and overlap conditions, making it promising for NISQ and pre-fault tolerant quantum hardware.

Sample-Based Krylov Quantum Diagonalization (SKQD) is a hybrid quantum–classical computational algorithm for the efficient approximation of ground-state energies and eigenstates of many-body quantum Hamiltonians, particularly suitable for near-term and pre-fault-tolerant quantum processors. SKQD synthesizes the systematic convergence properties of Krylov subspace approaches with the measurement efficiency of sample-based subspace construction, providing provable noise robustness and scalability under specific state structure and hardware assumptions (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025, Zhang et al., 2023, Lee et al., 4 Sep 2024, Byrne et al., 23 Dec 2024). The central innovation is the replacement of expensive matrix‐element estimation (e.g., via Hadamard tests) with computational-basis measurements of time-evolved quantum states, followed by classical diagonalization in the span of observed bitstrings, achieving eigenvalue estimates that converge under modest sparsity and overlap conditions.

1. Krylov Subspace Construction and Sampling Principles

SKQD leverages the notion that a short-time quantum evolution, starting from a judiciously chosen reference state $|\psi_0\rangle$, can generate a Krylov subspace that captures the essential spectral features of the Hamiltonian $H$. This subspace has the form

$$\mathcal{K}_d(H,|\psi_0\rangle) = \mathrm{span}\{\,|\psi_k\rangle = e^{-i k H \Delta t}|\psi_0\rangle \mid k = 0, \dots, d-1\,\}$$

with $d$ the Krylov dimension and $\Delta t$ a time step related to the spectral width, typically chosen as $\pi/\Delta E_{\rm max}$ (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025). Unlike standard Krylov Quantum Diagonalization (KQD), SKQD does not require full reconstruction of $\langle\psi_j|H|\psi_k\rangle$ or $\langle \psi_j|\psi_k\rangle$ via controlled-unitary circuits. Instead, for each Krylov state $|\psi_k\rangle$, SKQD performs $M$ computational-basis measurements, collecting distinct bitstrings $\{a_{k,m}\}_{m=1}^M$. The union of these configurations forms the sample-based subspace in which the Hamiltonian is projected and subsequently diagonalized classically.

Bitstrings observed with non-negligible probability (optionally postselected for enforcing physical symmetries) specify the computational basis $\{|b_i\rangle\}$ for the reduced subspace. Matrix elements $(H_{B})_{ij} = \langle b_i|H|b_j\rangle$ are computed directly from the classical description of $H$, e.g., via Pauli decomposition or fermionic integrals. The overlap matrix $S_B$ is the identity in the computational basis, obviating the need for further orthogonalization (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025).

2. Algorithmic Step-by-Step Structure and Classical Postprocessing

The SKQD algorithm is organized as follows (adapted from (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025, Lee et al., 4 Sep 2024)):

1. Reference State Preparation: Initialize $|\psi_0\rangle$ with sufficient overlap $|\langle\phi_0|\psi_0\rangle|^2 > 1/\text{poly}(n)$ with the target ground state $|\phi_0\rangle$.
2. Krylov State Generation: For $k=0, \dots, d-1$, prepare $$|\psi_k\rangle \approx (e^{-i H \Delta t})^k |\psi_0\rangle$$ using shallow Trotter or randomized qDRIFT circuits (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025).
3. Sampling: Measure all qubits in the $Z$ basis $M$ times per $|\psi_k\rangle$, aggregating observed bitstrings.
4. Subspace Formation: Enumerate the set of unique bitstrings to create the basis $\{|b_i\rangle\}$ for the sampled subspace.
5. Hamiltonian Projection: Compute the projected matrix $H_B$ with entries $\langle b_i|H|b_j\rangle$.
6. Diagonalization and Ground-State Estimation: Solve the eigenvalue problem $H_B w = E w$, take the smallest eigenvalue $\hat{E}$ as the ground-state energy estimate, and $|\tilde{\psi}\rangle = \sum_i w_i |b_i\rangle$ as the approximate ground state.

Efficient classical postprocessing, including configuration recovery and singular-value thresholding, is employed for mitigating noise, maintaining subspace conditioning, and controlling errors in the generalized eigenvalue problem (Rosanowski et al., 30 Oct 2025, Lee et al., 4 Sep 2024).

3. Convergence Guarantees and Sparsity Assumptions

The provable convergence of SKQD relies on state concentration and spectral assumptions. Let $|\phi_0\rangle$ be $(\alpha_L, \beta_L)$-sparse in the measurement basis: $\sum_{j=1}^L |g_j|^2 \ge \alpha_L$ and $|g_j|^2 \ge \beta_L$ for $j=1 \dots L$ with $L=\text{poly}(n)$ (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025). Provided nontrivial overlap $$|\gamma_0|^2 = |\langle\phi_0|\psi_0\rangle|^2 \gtrsim 1/\text{poly}(n)$$ and spectral gap $\Delta E_1 \gtrsim 1/\text{poly}(n)$, the key theorem states that sampling

$$M \ge \frac{d^2}{|\gamma_0|^2}\frac{1}{\beta_L} \log\left(\frac{L}{\eta}\right)$$

per Krylov state ensures recovery of all $L$ important basis configurations with probability $\ge 1 - \eta$. The resulting ground-state energy error satisfies

$$\hat{E} - E_0 \le \sqrt{8}\,\|H\| (1 - \sqrt{\alpha_L^{(0)}})^{1/2}$$

with total quantum cost $O(d M)$ and classical diagonalization cost $O((dM)^3)$ (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025).

Random sampling error contributions are analyzed via non-asymptotic random matrix perturbation bounds, which give explicit relationships between shot count, Krylov order, and error threshold, with strategies for controlling ill-conditioning through singular-value truncation (Zhang et al., 2023, Lee et al., 2023, Lee et al., 4 Sep 2024). When the ground-state wavefunction is well-concentrated, SKQD achieves systematic, polynomial-time convergence analogous to phase estimation, but with reduced circuit and measurement depth.

4. Practical Implementations and Sampling Error Mitigation

Beyond generic sampling protocols, SKQD performance critically depends on the measurement scheme for matrix elements. Two principal fragmentation strategies are reported: linear combination of unitaries (LCU) and diagonalizable fragments (FH/grouped Pauli). Associated variances and sample complexity bounds are derived to optimize measurement allocation (Lee et al., 4 Sep 2024).

Advanced sampling-error reduction includes:

• Shifting Technique: Introduction of a shift operator $\hat T$ to annihilate redundant Hamiltonian components, reducing the effective fragment norm $\|\bm\zeta(H-T)\|_1$ and thus sampling cost ($>20\times$ reduction) (Lee et al., 4 Sep 2024).
• Iterative Coefficient Splitting (ICS): Optimizes allocation of Pauli coefficients across measurement groups, minimizing total sample variance and achieving further factor-$2-5$ reduction, especially for electronic-structure Hamiltonians (Lee et al., 4 Sep 2024).
• Singular-Value Thresholding: Applies dimensionality reduction in the overlap matrix to control amplification of sampling noise (Lee et al., 2023, Lee et al., 4 Sep 2024).

Empirical testing on small molecular systems confirms up to $500\times$ sampling-cost reductions in $\ce{H2O}$, with errors concentrated within chemical accuracy for $M \sim 10^8$ shots rather than $10^{10}$–$10^{11}$ (Lee et al., 4 Sep 2024).

5. Comparative Performance and Numerical Benchmarks

Numerical investigations demonstrate SKQD's competitive performance on a variety of models:

• Transverse-Field Ising Model: Energy error lower than standard KQD at comparable shot costs, attributable to superior noise resilience in sample-based projection (Yu et al., 16 Jan 2025).
• Single-Impurity Anderson Model (SIAM): Largest ground-state quantum simulation ($85$ qubits, $41$ bath sites), with relative energy errors $10^{-5}-10^{-4}$ and agreement with DMRG calculations (Yu et al., 16 Jan 2025).
• Schwinger Model (Lattice Gauge Theory with $\theta$-term): Efficient resolution of phase transitions and reduction of Hilbert space dimensionality by $80\%$ ($N=20$ qubits, Krylov dimension $19\%$ of full sector); achievable accuracy $\sim 10^{-3}$ across hardware platforms (Rosanowski et al., 30 Oct 2025).

Scaling analysis demonstrates that while the Krylov subspace dimension grows exponentially, its reduced size compared to the full Hilbert space enables tractable classical diagonalization and paves the way for simulating larger quantum systems than previously possible by brute-force methods (Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025).

6. Advantages, Limitations, and Potential Extensions

Advantages:

• Shallow Circuit Depth: Only short-time dynamics are necessary, compatible with current NISQ and early-fault-tolerant hardware (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025).
• Shot Efficiency: Computational-basis sampling replaces Hadamard tests; measurement overhead is substantially reduced (Yu et al., 16 Jan 2025).
• Noise Robustness: Sample construction concentrates on important basis states, mitigating shot and hardware noise (Rosanowski et al., 30 Oct 2025).
• Classical Processing: Small-scale classical eigenproblem simplifies postprocessing and parallelization.

Limitations:

• Sparsity Assumption: Requires the target ground state to be sparse or well concentrated in the measurement basis; many quantum chemistry ground states may not comply (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025).
• Classical Scaling: The diagonalization cost grows rapidly with the number of samples; excessive measurement can lead to intractable $O((dM)^3)$ classical cost.
• Subspace Choice Sensitivity: Convergence critically depends on the choice of initial reference and time-step; parameter tuning is often required (Yu et al., 16 Jan 2025).

Extension Opportunities:

• Adaptive Sampling: Online pruning of inactive configurations and importance sampling during runtime (Yu et al., 16 Jan 2025, Lee et al., 4 Sep 2024).
• Hybrid Subspace Construction: Integration of Krylov states with other variational or mean-field states for richer approximation (Yu et al., 16 Jan 2025).
• Alternative Bases: Employing 1-RDM or Pauli-basis sampling to match sparsity patterns of the ground state (Yu et al., 16 Jan 2025, Piccinelli et al., 4 Aug 2025).
• Multistate Diagonalization: Extending subspace techniques to excited states, finite density, and higher-dimensional gauge theories (Rosanowski et al., 30 Oct 2025).

7. Implementation Protocols and Prospects for Quantum Hardware

SKQD is currently implemented on diverse quantum hardware architectures, including trapped-ion and superconducting platforms. It is characterized by:

• Preparation and Evolution: Reference state initialization followed by Trotterized, kinetic part or randomized qDRIFT unitary evolution (Yu et al., 16 Jan 2025, Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025).
• Measurement Strategy: Extensive computational-basis sampling, symmetry postselection, and error mitigation via configuration recovery (Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025).
• Scalability: Demonstrated convergence and accuracy up to $N=30$ qubits with Hilbert space reductions by several orders of magnitude (Rosanowski et al., 30 Oct 2025, Piccinelli et al., 4 Aug 2025).
• Resource Estimates: Sample complexity scales as $M = O(\|\bm\zeta_{\rm opt}\|_1^2/\epsilon^2)$ with fragment-norm optimization, yielding feasibility for EFTQC (Lee et al., 4 Sep 2024).
• Classical Postprocessing: Eigenvalue problem solution, basis truncation, coefficient optimization, and error analysis with polynomial-time scaling under favorable sparsity (Lee et al., 2023, Lee et al., 4 Sep 2024).

Future directions include development of enhanced error-mitigation protocols, systematic reference-state selection strategies, acceleration via qubitization or higher-order Trotterizations, and broadening to correlated electron systems and lattice gauge models (Rosanowski et al., 30 Oct 2025).

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Representative Papers:

• Quantum-Centric Algorithm for Sample-Based Krylov Diagonalization (Yu et al., 16 Jan 2025)
• Sample-Based Krylov Quantum Diagonalization for the Schwinger Model on Trapped-Ion and Superconducting Quantum Processors (Rosanowski et al., 30 Oct 2025)
• Quantum chemistry with provable convergence via randomized sample-based quantum diagonalization (Piccinelli et al., 4 Aug 2025)
• Measurement-efficient quantum Krylov subspace diagonalisation (Zhang et al., 2023)
• Efficient Strategies for Reducing Sampling Error in Quantum Krylov Subspace Diagonalization (Lee et al., 4 Sep 2024)
• A Quantum-Centric Super-Krylov Diagonalization Method (Byrne et al., 23 Dec 2024)