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arXiv:2607.09476

Orthogonal Quantum Krylov Diagonalisation

Hadi Rammal, Alexandre Perrin, Oumaya Ladhari, Clément Dutreix, Jérémie Messud, Matthieu Saubanere

quant-phSubmitted Jul 10, 2026• Announced Jul 13, 2026• 0 citations• Revised Jul 17, 2026 (v2)
v1 · Jul 10, 2026v2 · Jul 17, 2026
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Abstract

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 $J_1$--$J_2$ 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.

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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.
We present an algorithm that uses block encoding on a quantum computer to exactly construct a Krylov space, which can be used as the basis for the Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the classical Lanczos method has exponential cost in the system size to represent the Krylov states for quantum systems, our efficient quantum algorithm achieves this in polynomial time and memory. The construction presented is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise. This is possible because, unlike previous quantum Krylov methods, our algorithm does not require simulating real or imaginary time evolution. We provide an explicit error bound for the resulting ground state energy estimate in the presence of noise. For our method to be successful efficiently, the only requirement on the input problem is that the overlap of the initial state with the true ground state must be Ω ( 1 / poly ( n ) ) for n qubits.
Quantum Krylov subspace diagonalization (QKSD) is an emerging method used in place of quantum phase estimation in the early fault-tolerant era, where limited quantum circuit depth is available. In contrast to the classical Krylov subspace diagonalization (KSD) or the Lanczos method, QKSD exploits the quantum computer to efficiently estimate the eigenvalues of large-size Hamiltonians through a faster Krylov projection. However, unlike classical KSD, which is solely concerned with machine precision, QKSD is inherently accompanied by errors originating from a finite number of samples. Moreover, due to difficulty establishing an artificial orthogonal basis, ill-conditioning problems are often encountered, rendering the solution vulnerable to noise. In this work, we present a nonasymptotic theoretical framework to assess the relationship between sampling noise and its effects on eigenvalues. We also propose an optimal solution to cope with large condition numbers by eliminating the ill-conditioned bases. Numerical simulations of the one-dimensional Hubbard model demonstrate that the error bound of finite samplings accurately predicts the experimental errors in well-conditioned regions.
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 (QSD) with a Krylov space spanned by Chebyshev polynomials of the Hamiltonian is a natural ground-state method for fault-tolerant quantum computers, but every guarantee it offers---convergence rate, noise floor, and the cost of turning its classical output back into a physical state---is controlled by the overlap of the reference state with the true ground state, which decays exponentially with system size for any product state and non-trivial Hamiltonian. We propose quantum folding: for a system governed by a local Hamiltonian, seed the QSD with the product of the ground states of its subsystems, which differs from the target only through the interface terms. We prove a nonasymptotic error bound for the Chebyshev QSD that is linear in the measurement noise, by transplanting Kirby's effective-Hamiltonian analysis to the Chebyshev--Krylov basis, and identify the block-encoding rescaling of the Hamiltonian as a parameter of the guarantee. Exact simulations of antiferromagnetic Heisenberg chains show that the folded reference state reduces the Krylov dimension for a given accuracy severalfold and lowers the error under noise by two to five orders of magnitude relative to the Néel state. We show how the Ritz vector is prepared as a physical state and that the ground-state generation probability obeys a tight ceiling set by the overlap, so that recursive folding prepares ground states at polynomial cost. Taken together, these results allow for a scalable QSD framework that avoids the exponentially small ground-state overlaps associated with conventional product-state initialization. Finally we compute the fold fidelity itself: a slow power law for the gapless spin-1/2 chain, a size-independent constant for the gapped spin-111 chain, and an exact closed form converging to 1/4 for the AKLT chain.
Quantum Krylov subspace methods can extract ground and excited states by diagonalizing the Hamiltonian in a compact variational space. In practice, these spaces are almost always generated by real or imaginary time evolution, forcing a timestep trade-off between dynamical accuracy and basis collapse and often producing ill-conditioned overlap matrices that stall convergence. Here we introduce Quantum Krylov using Unitary Decomposition (QKUD), a time-evolution-free construction that maps Hamiltonian powers to implementable unitaries via the Hermitian transform sin⁡(εH)/ε\sin(εH)/εsin(εH)/ε. QKUD reduces to the exact Hamiltonian-power Krylov recursion as ε→0ε\rightarrow0ε→0, while finite εεε provides a controllable deformation that tunes subspace geometry and improves conditioning. Across molecular active-space benchmarks and a frustrated 2D J1-J2 Heisenberg model, QKUD reproduces exact-Krylov convergence in well-conditioned regimes and systematically restores variational improvement when both exact Krylov and time-evolution Krylov stagnate. These results identify overlap conditioning, instead of time-evolution fidelity, is the key resource for robust quantum Krylov simulation and provide a resilient way forward for accurate quantum simulation of challenging quantum many-body problems.
The Non-Orthogonal Quantum Eigensolver (NOQE) provides an accurate framework for electronic-structure calculations, but the estimation of its Hamiltonian and overlap matrix elements relies on sampling and requires O(1/ε2)O(1/\varepsilon^2)O(1/ε2) circuit repetitions to achieve additive precision ε\varepsilonε. Here, we reformulate this matrix-element estimation step as a collection of amplitude-estimation tasks and integrate iterative quantum amplitude estimation into the NOQE workflow. The resulting protocol achieves near-Heisenberg query complexity O(1/ε)O(1/\varepsilon)O(1/ε) for these estimation tasks, by replacing incoherent statistical averaging with coherent amplitude amplification. We present explicit circuit constructions and the corresponding implementation procedure. Numerical simulations for the electronic states of the hydrogen molecule show that the proposed method reaches chemical accuracy with substantially fewer total queries than the original sampling-based protocol. Overall, this work provides a measurement-efficient route to high-precision energy estimation and illustrates how sampling-limited quantum algorithms can be systematically reformulated to leverage quantum coherence and achieve lower measurement costs.