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arXiv:2607.23604

Performance and Stability of Quantum Krylov Diagonalization for the Hubbard Model

Mohammad Mirzakhani, Hyukgun Kwon, Kyungsun Moon

quant-phSubmitted Jul 26, 2026• Announced Jul 28, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum Krylov diagonalization (QKD) has emerged as a promising hybrid quantum-classical approach for estimating ground-state properties of many-body systems on near-term quantum devices. In this work, we investigate the convergence, stability, and hardware performance of QKD for the one-dimensional Hubbard model with periodic boundary conditions. Building upon our previously developed low-depth Jordan--Wigner implementation, which reduces the number of two-qubit (CNOT) gates required for quantum time evolution, we perform a systematic study of the influence of the Krylov dimension, Hamiltonian evolution parameters, system size, interaction strength, and singular-value truncation (SVT) on the convergence of the method. Our results show that the performance of QKD is governed by a delicate interplay between the low-energy spectral structure of the Hamiltonian and numerical stability. In particular, systems with near-closing energy gaps require longer evolution times to efficiently resolve nearby eigenstates, while the evolution time, Krylov dimension, Trotter number, and SVT threshold must be carefully balanced to avoid numerical instabilities and accumulated time-discretization errors. This analysis provides practical guidelines for selecting algorithmic parameters in QKD. Finally, we demonstrate the algorithm on IBM quantum hardware, where the experimental results reproduce the convergence trends predicted by ideal simulations using only lightweight readout-error mitigation and a modest measurement budget. Together, these results demonstrate that QKD is a practical and hardware-efficient approach for studying strongly correlated fermionic systems on current NISQ quantum processors.

Comments
15 pages, 12 figures

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Quantum Krylov diagonalization (QKD) has emerged as a promising hybrid quantum-classical approach for estimating ground-state properties of many-body systems on near-term quantum devices. In this work, we investigate the convergence, stability, and hardware performance of QKD for the one-dimensional Hubbard model with periodic boundary conditions. Building upon our previously developed low-depth Jordan--Wigner implementation, which reduces the number of two-qubit (CNOT) gates required for quantum time evolution, we perform a systematic study of the influence of the Krylov dimension, Hamiltonian evolution parameters, system size, interaction strength, and singular-value truncation (SVT) on the convergence of the method. Our results show that the performance of QKD is governed by a delicate interplay between the low-energy spectral structure of the Hamiltonian and numerical stability. In particular, systems with near-closing energy gaps require longer evolution times to efficiently resolve nearby eigenstates, while the evolution time, Krylov dimension, Trotter number, and SVT threshold must be carefully balanced to avoid numerical instabilities and accumulated time-discretization errors. This analysis provides practical guidelines for selecting algorithmic parameters in QKD. Finally, we demonstrate the algorithm on IBM quantum hardware, where the experimental results reproduce the convergence trends predicted by ideal simulations using only lightweight readout-error mitigation and a modest measurement budget. Together, these results demonstrate that QKD is a practical and hardware-efficient approach for studying strongly correlated fermionic systems on current NISQ quantum processors.
Quantum subspace-diagonalization methods, particularly Quantum Krylov Diagonalization (QKD), provide a promising route for computing low-energy spectra of quantum many-body Hamiltonians. However, existing quantum Krylov approaches rely on non-orthogonal Krylov bases, requiring overlap-matrix regularization that limits numerical stability and accuracy. In this work, we introduce an Orthogonal Quantum Krylov Diagonalization (OQKD) framework that reformulates the classical Lanczos recursion at the operator level, enabling an orthogonal quantum implementation of Krylov-subspace diagonalization. By expressing Lanczos vectors as polynomial transformations of the Hamiltonian, OQKD reproduces the orthogonality, tridiagonal structure, and convergence behavior of the classical Lanczos algorithm thus eliminating the need for overlap-matrix regularization. We further show that the required Lanczos polynomials can be implemented using block encoding and Generalized Quantum Signal Processing with the same asymptotic query complexity as Chebyshev-based QKD methods. Numerical simulations of the J1J_1J1​--J2J_2J2​ Heisenberg model confirm the classical Lanczos convergence and numerical stability of the proposed method, while the measurement-complexity scaling is established analytically. Building upon the OQKD framework, we then introduce a restarted state-preparation protocol that replaces a single high-degree polynomial transformation with a sequence of fixed low-degree transformations, maintaining an affordable block encoding success probability while retaining comparable convergence. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state-preparation strategy for Quantum Phase Estimation.
The estimation of low energies of many-body systems is a cornerstone of the computational quantum sciences. Variational quantum algorithms can be used to prepare ground states on pre-fault-tolerant quantum processors, but their lack of convergence guarantees and impractical number of cost function estimations prevent systematic scaling of experiments to large systems. Alternatives to variational approaches are needed for large-scale experiments on pre-fault-tolerant devices. Here, we use a superconducting quantum processor to compute eigenenergies of quantum many-body systems on two-dimensional lattices of up to 56 sites, using the Krylov quantum diagonalization algorithm, an analog of the well-known classical diagonalization technique. We construct subspaces of the many-body Hilbert space using Trotterized unitary evolutions executed on the quantum processor, and classically diagonalize many-body interacting Hamiltonians within those subspaces. These experiments demonstrate exponential convergence towards an estimate of the ground state energy, and show that quantum diagonalization algorithms are poised to complement their classical counterparts at the foundation of computational methods for quantum systems. The estimation of low energies of many-body systems is a cornerstone of the computational quantum sciences. This paper demonstrates on a superconducting quantum processor that the Krylov quantum diagonalization algorithm is poised to complement its classical counterparts at the foundation of computational methods for quantum systems.
Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.
Sample-based Krylov quantum diagonalization (SKQD) is a paradigmatic example of a quantum-centric supercomputing workflow that combines the convergence structure of Krylov quantum diagonalization with classical sampling-based post-processing. It provides convergence guarantees assuming that the important computational-basis configurations can be sampled from a set of Krylov states with sufficient probability. We analyze this assumption under depolarizing noise, deriving shot-count resource estimates that expose an exponential depth penalty, and perform experiments on current noisy hardware with the one-dimensional single-impurity Anderson model on a 20-site (40-qubit) instance. Device noise breaks the practical convergence predicted by the noiseless SKQD analysis, but approximate compilation techniques can compress the Krylov time-evolution circuits before execution. The compressed circuits accumulate less hardware noise, recover the expected energy convergence, and reduce both the quantum shot budget and the classical subspace dimension, remaining beneficial even when classical configuration recovery is applied.
We propose a class of randomized quantum Krylov diagonalization (rQKD) algorithms capable of solving the eigenstate estimation problem with modest quantum resource requirements. Compared to previous real-time evolution quantum Krylov subspace methods, our approach expresses the time evolution operator ${e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\tau}}$ as a linear combination of unitaries and subsequently uses a stochastic sampling procedure to reduce circuit depth requirements. While our methodology applies to any Hamiltonian with fast-forwardable subcomponents, we focus on its application to the explicitly double-factorized electronic-structure Hamiltonian. To demonstrate the potential of the proposed rQKD algorithm on near-term quantum devices, we provide numerical benchmarks for a variety of molecular systems with circuit-based state-vector simulators including the effects of sampling noise, achieving ground-state energy errors of less than $1 \mathrm{kcal} {\mathrm{mol}}^{\ensuremath{-}1}$ with circuit depths orders of magnitude shallower than those required for low-rank deterministic Trotter-Suzuki decompositions.
Quantum Krylov algorithms are a promising approach for ground-state energy estimation on near-term quantum computers. However, their sampling cost is inherently high because each Hamiltonian term is measured individually. Despite various mitigation techniques, this overhead remains a bottleneck for large-scale electronic structure problems. We introduce mirror subspace diagonalization (MSD), which approaches the theoretical lower bound on the sampling cost for quantum Krylov algorithms built upon real-time evolution. MSD uses a finite-difference formula to express the Hamiltonian as a linear combination of time-evolution unitaries with symmetrically shifted timesteps, enabling efficient estimation of the Hamiltonian matrix within the Krylov subspace. By optimizing the timestep and shifting the energy spectrum, finite-difference and statistical errors are minimized simultaneously, allowing MSD to attain the sampling-cost lower bound up to a logarithmic factor. We further apply classical post-processing to infer Hamiltonian moments, which mitigate the ground-state energy error via the Lanczos scheme. Our analysis shows MSD is particularly effective when the Hamiltonian's spectral norm is much smaller than its 1-norm, as in high-accuracy molecular simulations with large basis sets and strong electronic correlations. Numerical results for various molecular models show sampling-cost reductions of roughly 10 to 10,000 times over the conventional quantum Krylov algorithm.