Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2506.09971 · Physical Review A

Quantum block Krylov subspace projector algorithm for computing low-lying eigenenergies

Maria Gabriela Jordão Oliveira, Nina Glaser

Published Nov 20, 2025• 0 citations
arXivHTMLPDF

Abstract

Computing eigenvalues is a computationally intensive task central to many applications in the natural sciences, particularly in quantum physics and chemistry, where determining low-lying energy spectra is key to characterizing physical and chemical properties. Toward this end, we present and investigate the quantum block Krylov subspace projector (QBKSP) algorithm---a multireference quantum Lanczos method---to accurately compute low-lying eigenenergies, including degenerate ones. We propose three compact quantum circuits, each suited to different problem settings, for evaluating the necessary expectation values. In contrast to previous multireference quantum Krylov methods, QBKSP achieves linear scaling with respect to the number of Krylov iterations instead of a quadratic one, and furthermore reduces the scaling with respect to the number of references by half for real Hamiltonians and reference states. To assess the impact of the number and fidelity of the initial reference states, as well as the QBKSP performance in different parameter regimes, we perform both error-free and sampling noise-limited simulations. Our results demonstrate that employing multiple reference states significantly improves convergence, particularly in noisy scenarios or when a single reference fails to capture all target eigenstates. Moreover, the QBKSP algorithm allows for the determination of degenerate eigenstates and their multiplicities through appropriate convergence conditions.

DOI
10.1103/p8wj-gplb

Related papers

Multi-reference Quantum Davidson Algorithm for Quantum Dynamics

Noah Berthusen, Alam, Faisal, Yu Zhang

Simulating quantum systems is one of the most promising tasks where quantum computing can potentially outperform classical computing. However, the robustness needed for reliable simulations of medium to large systems is beyond the reach of existing quantum devices. To address this, Quantum Krylov Subspace (QKS) methods have been developed, enhancing the ability to perform accelerated simulations on noisy intermediate-scale quantum computers. In this study, we introduce and evaluate two QKS methods derived from the QDavidson algorithm, a novel approach for determining the ground and excited states of many-body systems. Unlike other QKS methods that pre-generate the Krylov subspace through real- or imaginary-time evolution, QDavidson iteratively adds basis vectors into the Krylov subspace. This iterative process enables faster convergence with fewer iterations and necessitates shallower circuit depths, marking a significant advancement in the field of quantum simulation.

arXiv:2406.08675•Jun 12, 2024•0 citations

Measurement-efficient quantum Krylov subspace diagonalisation

Zongkang Zhang, Anbang Wang, Xiaosi Xu +1 more

The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are prone to errors due to inevitable statistical fluctuations in quantum measurements. To address this problem, we develop a general theoretical framework to analyse the statistical error and measurement cost. Based on the framework, we propose a quantum algorithm to construct the Hamiltonian-power Krylov subspace that can minimise the measurement cost. In our algorithm, the product of power and Gaussian functions of the Hamiltonian is expressed as an integral of the real-time evolution, such that it can be evaluated on a quantum computer. We compare our algorithm with other established quantum Krylov subspace algorithms in solving two prominent examples. To achieve an error comparable to that of the classical Lanczos algorithm at the same subspace dimension, our algorithm typically requires orders of magnitude fewer measurements than others. Such an improvement can be attributed to the reduced cost of composing projectors onto the ground state. These results show that our algorithm is exceptionally robust to statistical fluctuations and promising for practical applications.

arXiv:2301.13353•Quantum•Aug 13, 2024•15 citations

Nuclear Spectra from Quantum Lanczos Algorithm with Real-Time Evolution and Multiple Reference States

Bowman, Amanda

Models of quantum systems scale exponentially with the addition of single-particle states, which can present computationally intractable problems. Alternatively, quantum computers can store a many-body basis of $2^n$ dimensions on $n$ qubits. This motivated the quantum eigensolver algorithms developed in recent years, such as the quantum Lanczos algorithm based on the classical, iterative Lanczos algorithm. I performed numerical simulations to find the low-lying eigenstates of $^{20}$Ne, $^{22}$Na, and $^{29}$Na to compare imaginary- and real-time evolution. Though imaginary-time evolution leads to faster convergence, real-time evolution still converges within tens of iterations and satisfies the requirement for unitary operators on quantum computers. Additionally, using multiple reference states leads to faster convergences or higher accuracy for a fixed number of real-time iterations. I performed quantum circuit prototype numerical simulations on a classical computer of the QLanczos algorithm with real-time evolution and multiple reference states to find the low-lying eigenstates of $^{8}$Be. These simulations were run in both the spherical basis and Hartree-Fock basis, demonstrating that an M-scheme spherical basis leads to lower depth circuits than the Hartree-Fock basis. Finally, I present the quantum circuits for the QLanczos algorithm with real-time evolution and multiple references.

arXiv:2309.00759•Sep 1, 2023•0 citations

Exact and efficient Lanczos method on a quantum computer

William Kirby, Mário Motta, Antonio Mezzacapo

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.

arXiv:2208.00567•Quantum•May 23, 2023•66 citations

Generative Krylov Subspace Representations for Scalable Quantum Eigensolvers

Changwon Lee, Daniel K. Park

Predicting ground state energies of quantum many-body systems is one of the central computational challenges in quantum chemistry, physics, and materials science. Krylov subspace methods, such as Krylov Quantum Diagonalization and Sample-based Krylov Quantum Diagonalization, are promising approaches for this task on near-term quantum computers. However, both require repeated quantum circuit executions for each Krylov subspace and for every new Hamiltonian, posing a major bottleneck under noisy hardware constraints. We introduce Generative Krylov Subspace Representations (GenKSR), a framework that learns a classical generative representation of the entire Krylov diagonalization process. To enable effective modeling of quantum systems, GenKSR leverages a conditional generative model framework. We investigate two representative backbone architectures, the standard Transformer and the Mamba state-space model. By learning the distribution of measurement outcomes conditioned on Hamiltonian parameters and evolution time, GenKSR generates Krylov subspace samples for unseen Hamiltonians and for larger subspace dimensions than those used in training. This enables full energy reconstruction purely from the classical model, without additional quantum experiments. We validate our approach through simulations of 15-qubit 1D and 16-qubit 2D Heisenberg models, as well as a hardware experiment on a 20-qubit XXZ chain executed on an IBM quantum processor. Our model successfully learns the distribution from experimental data and generates a high-fidelity representation of the quantum process. This representation enables classical reproduction of experimental outcomes, supports reliable energy estimates for unseen Hamiltonians, and significantly reduces the need for further quantum computation.

quant-pharXiv:2512.19420•Dec 23, 2025•0 citations

Sampling Error Analysis in Quantum Krylov Subspace Diagonalization

Gwonhak Lee, Dongkeun Lee, Joonsuk Huh

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.

arXiv:2307.16279•Quantum•Sep 19, 2024•15 citations
Computing eigenvalues is a computationally intensive task central to many applications in the natural sciences, particularly in quantum physics and chemistry, where determining low-lying energy spectra is key to characterizing physical and chemical properties. Toward this end, we present and investigate the quantum block Krylov subspace projector (QBKSP) algorithm---a multireference quantum Lanczos method---to accurately compute low-lying eigenenergies, including degenerate ones. We propose three compact quantum circuits, each suited to different problem settings, for evaluating the necessary expectation values. In contrast to previous multireference quantum Krylov methods, QBKSP achieves linear scaling with respect to the number of Krylov iterations instead of a quadratic one, and furthermore reduces the scaling with respect to the number of references by half for real Hamiltonians and reference states. To assess the impact of the number and fidelity of the initial reference states, as well as the QBKSP performance in different parameter regimes, we perform both error-free and sampling noise-limited simulations. Our results demonstrate that employing multiple reference states significantly improves convergence, particularly in noisy scenarios or when a single reference fails to capture all target eigenstates. Moreover, the QBKSP algorithm allows for the determination of degenerate eigenstates and their multiplicities through appropriate convergence conditions.
Simulating quantum systems is one of the most promising tasks where quantum computing can potentially outperform classical computing. However, the robustness needed for reliable simulations of medium to large systems is beyond the reach of existing quantum devices. To address this, Quantum Krylov Subspace (QKS) methods have been developed, enhancing the ability to perform accelerated simulations on noisy intermediate-scale quantum computers. In this study, we introduce and evaluate two QKS methods derived from the QDavidson algorithm, a novel approach for determining the ground and excited states of many-body systems. Unlike other QKS methods that pre-generate the Krylov subspace through real- or imaginary-time evolution, QDavidson iteratively adds basis vectors into the Krylov subspace. This iterative process enables faster convergence with fewer iterations and necessitates shallower circuit depths, marking a significant advancement in the field of quantum simulation.
The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are prone to errors due to inevitable statistical fluctuations in quantum measurements. To address this problem, we develop a general theoretical framework to analyse the statistical error and measurement cost. Based on the framework, we propose a quantum algorithm to construct the Hamiltonian-power Krylov subspace that can minimise the measurement cost. In our algorithm, the product of power and Gaussian functions of the Hamiltonian is expressed as an integral of the real-time evolution, such that it can be evaluated on a quantum computer. We compare our algorithm with other established quantum Krylov subspace algorithms in solving two prominent examples. To achieve an error comparable to that of the classical Lanczos algorithm at the same subspace dimension, our algorithm typically requires orders of magnitude fewer measurements than others. Such an improvement can be attributed to the reduced cost of composing projectors onto the ground state. These results show that our algorithm is exceptionally robust to statistical fluctuations and promising for practical applications.
Models of quantum systems scale exponentially with the addition of single-particle states, which can present computationally intractable problems. Alternatively, quantum computers can store a many-body basis of 2n2^n2n dimensions on nnn qubits. This motivated the quantum eigensolver algorithms developed in recent years, such as the quantum Lanczos algorithm based on the classical, iterative Lanczos algorithm. I performed numerical simulations to find the low-lying eigenstates of 20^{20}20Ne, 22^{22}22Na, and 29^{29}29Na to compare imaginary- and real-time evolution. Though imaginary-time evolution leads to faster convergence, real-time evolution still converges within tens of iterations and satisfies the requirement for unitary operators on quantum computers. Additionally, using multiple reference states leads to faster convergences or higher accuracy for a fixed number of real-time iterations. I performed quantum circuit prototype numerical simulations on a classical computer of the QLanczos algorithm with real-time evolution and multiple reference states to find the low-lying eigenstates of 8^{8}8Be. These simulations were run in both the spherical basis and Hartree-Fock basis, demonstrating that an M-scheme spherical basis leads to lower depth circuits than the Hartree-Fock basis. Finally, I present the quantum circuits for the QLanczos algorithm with real-time evolution and multiple references.
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.
Predicting ground state energies of quantum many-body systems is one of the central computational challenges in quantum chemistry, physics, and materials science. Krylov subspace methods, such as Krylov Quantum Diagonalization and Sample-based Krylov Quantum Diagonalization, are promising approaches for this task on near-term quantum computers. However, both require repeated quantum circuit executions for each Krylov subspace and for every new Hamiltonian, posing a major bottleneck under noisy hardware constraints. We introduce Generative Krylov Subspace Representations (GenKSR), a framework that learns a classical generative representation of the entire Krylov diagonalization process. To enable effective modeling of quantum systems, GenKSR leverages a conditional generative model framework. We investigate two representative backbone architectures, the standard Transformer and the Mamba state-space model. By learning the distribution of measurement outcomes conditioned on Hamiltonian parameters and evolution time, GenKSR generates Krylov subspace samples for unseen Hamiltonians and for larger subspace dimensions than those used in training. This enables full energy reconstruction purely from the classical model, without additional quantum experiments. We validate our approach through simulations of 15-qubit 1D and 16-qubit 2D Heisenberg models, as well as a hardware experiment on a 20-qubit XXZ chain executed on an IBM quantum processor. Our model successfully learns the distribution from experimental data and generates a high-fidelity representation of the quantum process. This representation enables classical reproduction of experimental outcomes, supports reliable energy estimates for unseen Hamiltonians, and significantly reduces the need for further quantum computation.
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.