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:2605.11787

CERTIFY-ED: A Multi-Layer Verification Framework for Exact Diagonalization of Quantum Many-Body Systems

Sarang Vehale, Ritu Goel

Published May 12, 2026• 0 citations
arXivHTMLPDF

Abstract

Exact diagonalization (ED) is a workhorse technique in computational quantum many-body physics, but published ED results are rarely accompanied by machine-checkable evidence of their numerical correctness. The community typically relies on the implicit trust chain LAPACK $\to$ user code $\to$ result, with at most informal agreement against another package treated as confirmation. We argue that this practice is inadequate for a method whose output frequently underpins theoretical claims, and we present \textsc{certify-ed}, a verification framework designed to be used \emph{alongside} existing ED packages (QuSpin, XDiag, ALPS) rather than as a replacement for them. The framework consists of (i) a multi-oracle eigensolver that runs three independent LAPACK paths and reports their pairwise disagreement, (ii) thirteen logically independent validation layers covering algebraic invariants, analytic limits, alternative algorithms, arbitrary-precision reference computation, conservation laws, dynamical consistency, and finite-size scaling, and (iii) tamper-evident SHA-256 hashed certificates that downstream consumers can verify. The framework also ships an error-injection layer that confirms the entire pipeline detects six injected error classes. Running on sixteen physics models from one-dimensional spin chains to two-dimensional Kitaev honeycomb clusters, our reference implementation passes 53 of 53 unit tests and 81 of 81 individual validation tests in under thirty seconds, with maximum disagreement against QuSpin of $1.6\times 10^{-14}$ across 320 eigenvalue comparisons, and agreement with 50-digit \texttt{mpmath} reference values to $1.6\times 10^{-15}$. The package is released under the MIT license on Zenodo and Github

DOI
10.48550/arxiv.2605.11787

Related papers

Codebase release 0.4 for XDiag

Alexander Wietek, Luke Staszewski, Martin Ulaga +6 more

Exact diagonalization (ED) is a cornerstone technique in quantum many-body physics, enabling precise solutions to the many-body Schrödinger equation for interacting quantum systems. Despite its utility in studying ground states, excited states, and dynamical behaviors, the exponential growth of the Hilbert space with system size presents significant computational challenges. We introduce XDiag, an open-source software package designed to combine advanced and efficient algorithms for ED with and without symmetry-adapted bases with user-friendly interfaces. Implemented in C++ for computational efficiency and wrapped in Julia for ease of use, XDiag provides a comprehensive toolkit for ED calculations. Key features of XDiag include the first publicly accessible implementation of sublattice coding algorithms for large-scale spin system diagonalizations, efficient Lin table algorithms for symmetry lookups, and random-hashing techniques for distributed memory parallelization. The library supports various Hilbert space types (e.g., spin-1/2, electron, and t-J models), facilitates symmetry-adapted block calculations, and automates symmetry considerations. The package is complemented by extensive documentation, a user guide, reproducible benchmarks demonstrating near-linear scaling on thousands of CPU cores, and over 20 examples covering ground-state calculations, spectral functions, time evolution, and thermal states. By integrating high-performance computing with accessible scripting capabilities, XDiag allows researchers to perform state-of-the-art ED simulations and explore quantum many-body phenomena with unprecedented flexibility and efficiency.

SciPost Physics Codebases•May 7, 2026•0 citations

XDiag: Exact diagonalization for quantum many-body systems

Alexander Wietek, Luke Staszewski, Martin Ulaga +6 more

Exact diagonalization (ED) is a cornerstone technique in quantum many-body physics, enabling precise solutions to the many-body Schrödinger equation for interacting quantum systems. Despite its utility in studying ground states, excited states, and dynamical behaviors, the exponential growth of the Hilbert space with system size presents significant computational challenges. We introduce XDiag, an open-source software package designed to combine advanced and efficient algorithms for ED with and without symmetry-adapted bases with user-friendly interfaces. Implemented in C++ for computational efficiency and wrapped in Julia for ease of use, XDiag provides a comprehensive toolkit for ED calculations. Key features of XDiag include the first publicly accessible implementation of sublattice coding algorithms for large-scale spin system diagonalizations, efficient Lin table algorithms for symmetry lookups, and random-hashing techniques for distributed memory parallelization. The library supports various Hilbert space types (e.g., spin-1/2, electron, and t-J models), facilitates symmetry-adapted block calculations, and automates symmetry considerations. The package is complemented by extensive documentation, a user guide, reproducible benchmarks demonstrating near-linear scaling on thousands of CPU cores, and over 20 examples covering ground-state calculations, spectral functions, time evolution, and thermal states. By integrating high-performance computing with accessible scripting capabilities, XDiag allows researchers to perform state-of-the-art ED simulations and explore quantum many-body phenomena with unprecedented flexibility and efficiency.

arXiv:2505.02901•SciPost Physics Codebases•May 7, 2026•5 citations

Exact diagonalization of the one-dimensional hubbard model up to sixteen sites at half-filling

A. S. Olatinwo, N. Salako, A. Amosun

Numerical exact diagonalisation (ED) of the Hubbard model is traditionally constrained by the exponential growth of the Hilbert space. In this study, we implemented a symmetry-based ED framework to investigate the one-dimensional Hubbard model at half-filling for system sizes up to L=16. By exploiting particle number conservation, spin projection, and translation invariance, we significantly reduced the Hamiltonian dimension, enabling the resolution of ground state energies and dispersion relations without additional approximations. Our results confirmed the transition from a metallic to a correlated insulating regime with increasing on-site repulsion U. Furthermore, extending calculations from ten to sixteen sites revealed scaling trends that asymptotically approached the thermodynamic limit predicted by the Bethe Ansatz (𝐿= ∞). These results served as precise, reproducible benchmarks for validating approximate many-body techniques such as the Density Matrix Renormalisation Group (DMRG).

Ife Journal of Science•Jan 1, 2026•0 citations

lattice-symmetries: A package for working with quantum many-body bases

Tom Westerhout

Exact diagonalization (ED) is one of the most reliable and established numerical methods of quantum many-body theory.It is precise, unbiased, and general enough to be applicable to a huge variety of problems in condensed matter physics.Mathematically, ED is a linear algebra problem involving a matrix called the Hamiltonian.For a system of spin-1/2 particles, the size of this matrix scales exponentially (as O(2 N )) with the number of particles N .Very fast scaling of memory requirements with system size is the main computational challenge of the method.There are a few techniques allowing one to lower the amount of storage used by the Hamiltonian.For example, one can store only the non-zero elements of the Hamiltonian.This is beneficial when the Hamiltonian is sparse, which is usually the case in condensed matter physics.One can even take it one step further and avoid storing the matrix altogether by instead computing matrix elements on the fly.

arXiv:2104.04011•The Journal of Open Source Software•Aug 26, 2021•17 citations

Reproducibility Package — Conditional Entropy-Flow Bound & Ladder Benchmarks for Many-Body Localization (MBL)

Zeraoulia Rafik

This Zenodo record provides an independent, self-contained reproducibility package for the figures and tables reported in the accompanying manuscript on conditional entanglement bounds and ladder exact-diagonalization (ED) benchmarks for many-body localization. Contents Google Colab notebook (MBL_entropyflow_zenodo.ipynb) that reproduces: fig_entropy_flow.png (boundary-normalized mid-spectrum entanglement vs disorder strength) fig_drift.png (finite-size drift indicator Δ(W)) summary tables exported as CSV (entanglement and drift) Archived datasets (entropy_data_new.csv, and optional quench outputs if present) used to generate the figures and tables without rerunning ED. Optional ED script (compute_data.py) for regenerating datasets from scratch (computationally heavier; intended for users who wish to reproduce raw data). What the notebook reproduces Running the notebook generates: Figures in outputs/figures/ Tables in outputs/tables/ (CSV format) including: entropy_summary.csv drift_summary.csv Computational notes The notebook is designed to run end-to-end in Google Colab. Recomputing raw ED data can be time-consuming; the included CSVs allow immediate regeneration of the paper figures/tables. Results depend on system size (e.g., 4×2 and 5×2 ladders), disorder type (random vs quasiperiodic), and sampling parameters documented in the notebook. License Choose one of the following (recommended): MIT License (best for code + data reuse), or CC BY 4.0 (best if you want strict attribution for all materials). Citation If you use this package, please cite: The Zenodo DOI for this record (automatically generated by Zenodo), And the associated manuscript/preprint (add arXiv or journal reference here once available).

•Mar 2, 2026•0 citations

Certifying quantum states without independence assumptions

Mariana Navarro, Leonardo Zambrano

Standard quantum verification and certification protocols often assume that experimental sources emit independent and identically distributed (i.i.d.) states. In realistic scenarios, however, temporal drift, memory effects, feedback, and correlated noise can violate this assumption, causing standard analyses to underestimate uncertainty and overestimate device performance. Here, we introduce a general framework for quantum verification and certification in which the state produced in each round may depend arbitrarily on the preceding experimental history. Our method provides rigorous confidence intervals for observable expectation values averaged over the experimental trajectory. The framework covers both full verification, in which every state is measured, and spot-checking certification, in which a randomly selected subset is tested to certify the average observable value of the complementary, remaining systems. We apply the framework to importance-sampled Pauli measurements for direct fidelity estimation, energy estimation, and entanglement witnessing, and to classical shadows, allowing multiple observables to be certified simultaneously from the same test data. We illustrate the framework numerically and demonstrate it experimentally through Bell-state certification on a quantum processor.

quant-pharXiv:2606.31913•Jul 1, 2026•0 citations
Exact diagonalization (ED) is a workhorse technique in computational quantum many-body physics, but published ED results are rarely accompanied by machine-checkable evidence of their numerical correctness. The community typically relies on the implicit trust chain LAPACK →\to→ user code →\to→ result, with at most informal agreement against another package treated as confirmation. We argue that this practice is inadequate for a method whose output frequently underpins theoretical claims, and we present \textsc{certify-ed}, a verification framework designed to be used \emph{alongside} existing ED packages (QuSpin, XDiag, ALPS) rather than as a replacement for them. The framework consists of (i) a multi-oracle eigensolver that runs three independent LAPACK paths and reports their pairwise disagreement, (ii) thirteen logically independent validation layers covering algebraic invariants, analytic limits, alternative algorithms, arbitrary-precision reference computation, conservation laws, dynamical consistency, and finite-size scaling, and (iii) tamper-evident SHA-256 hashed certificates that downstream consumers can verify. The framework also ships an error-injection layer that confirms the entire pipeline detects six injected error classes. Running on sixteen physics models from one-dimensional spin chains to two-dimensional Kitaev honeycomb clusters, our reference implementation passes 53 of 53 unit tests and 81 of 81 individual validation tests in under thirty seconds, with maximum disagreement against QuSpin of 1.6×10−141.6\times 10^{-14}1.6×10−14 across 320 eigenvalue comparisons, and agreement with 50-digit \texttt{mpmath} reference values to 1.6×10−151.6\times 10^{-15}1.6×10−15. The package is released under the MIT license on Zenodo and Github
Exact diagonalization (ED) is a cornerstone technique in quantum many-body physics, enabling precise solutions to the many-body Schrödinger equation for interacting quantum systems. Despite its utility in studying ground states, excited states, and dynamical behaviors, the exponential growth of the Hilbert space with system size presents significant computational challenges. We introduce XDiag, an open-source software package designed to combine advanced and efficient algorithms for ED with and without symmetry-adapted bases with user-friendly interfaces. Implemented in C++ for computational efficiency and wrapped in Julia for ease of use, XDiag provides a comprehensive toolkit for ED calculations. Key features of XDiag include the first publicly accessible implementation of sublattice coding algorithms for large-scale spin system diagonalizations, efficient Lin table algorithms for symmetry lookups, and random-hashing techniques for distributed memory parallelization. The library supports various Hilbert space types (e.g., spin-1/2, electron, and t-J models), facilitates symmetry-adapted block calculations, and automates symmetry considerations. The package is complemented by extensive documentation, a user guide, reproducible benchmarks demonstrating near-linear scaling on thousands of CPU cores, and over 20 examples covering ground-state calculations, spectral functions, time evolution, and thermal states. By integrating high-performance computing with accessible scripting capabilities, XDiag allows researchers to perform state-of-the-art ED simulations and explore quantum many-body phenomena with unprecedented flexibility and efficiency.
Exact diagonalization (ED) is a cornerstone technique in quantum many-body physics, enabling precise solutions to the many-body Schrödinger equation for interacting quantum systems. Despite its utility in studying ground states, excited states, and dynamical behaviors, the exponential growth of the Hilbert space with system size presents significant computational challenges. We introduce XDiag, an open-source software package designed to combine advanced and efficient algorithms for ED with and without symmetry-adapted bases with user-friendly interfaces. Implemented in C++ for computational efficiency and wrapped in Julia for ease of use, XDiag provides a comprehensive toolkit for ED calculations. Key features of XDiag include the first publicly accessible implementation of sublattice coding algorithms for large-scale spin system diagonalizations, efficient Lin table algorithms for symmetry lookups, and random-hashing techniques for distributed memory parallelization. The library supports various Hilbert space types (e.g., spin-1/2, electron, and t-J models), facilitates symmetry-adapted block calculations, and automates symmetry considerations. The package is complemented by extensive documentation, a user guide, reproducible benchmarks demonstrating near-linear scaling on thousands of CPU cores, and over 20 examples covering ground-state calculations, spectral functions, time evolution, and thermal states. By integrating high-performance computing with accessible scripting capabilities, XDiag allows researchers to perform state-of-the-art ED simulations and explore quantum many-body phenomena with unprecedented flexibility and efficiency.
Numerical exact diagonalisation (ED) of the Hubbard model is traditionally constrained by the exponential growth of the Hilbert space. In this study, we implemented a symmetry-based ED framework to investigate the one-dimensional Hubbard model at half-filling for system sizes up to L=16. By exploiting particle number conservation, spin projection, and translation invariance, we significantly reduced the Hamiltonian dimension, enabling the resolution of ground state energies and dispersion relations without additional approximations. Our results confirmed the transition from a metallic to a correlated insulating regime with increasing on-site repulsion U. Furthermore, extending calculations from ten to sixteen sites revealed scaling trends that asymptotically approached the thermodynamic limit predicted by the Bethe Ansatz (𝐿= ∞). These results served as precise, reproducible benchmarks for validating approximate many-body techniques such as the Density Matrix Renormalisation Group (DMRG).
Exact diagonalization (ED) is one of the most reliable and established numerical methods of quantum many-body theory.It is precise, unbiased, and general enough to be applicable to a huge variety of problems in condensed matter physics.Mathematically, ED is a linear algebra problem involving a matrix called the Hamiltonian.For a system of spin-1/2 particles, the size of this matrix scales exponentially (as O(2 N )) with the number of particles N .Very fast scaling of memory requirements with system size is the main computational challenge of the method.There are a few techniques allowing one to lower the amount of storage used by the Hamiltonian.For example, one can store only the non-zero elements of the Hamiltonian.This is beneficial when the Hamiltonian is sparse, which is usually the case in condensed matter physics.One can even take it one step further and avoid storing the matrix altogether by instead computing matrix elements on the fly.
This Zenodo record provides an independent, self-contained reproducibility package for the figures and tables reported in the accompanying manuscript on conditional entanglement bounds and ladder exact-diagonalization (ED) benchmarks for many-body localization. Contents Google Colab notebook (MBL_entropyflow_zenodo.ipynb) that reproduces: fig_entropy_flow.png (boundary-normalized mid-spectrum entanglement vs disorder strength) fig_drift.png (finite-size drift indicator Δ(W)) summary tables exported as CSV (entanglement and drift) Archived datasets (entropy_data_new.csv, and optional quench outputs if present) used to generate the figures and tables without rerunning ED. Optional ED script (compute_data.py) for regenerating datasets from scratch (computationally heavier; intended for users who wish to reproduce raw data). What the notebook reproduces Running the notebook generates: Figures in outputs/figures/ Tables in outputs/tables/ (CSV format) including: entropy_summary.csv drift_summary.csv Computational notes The notebook is designed to run end-to-end in Google Colab. Recomputing raw ED data can be time-consuming; the included CSVs allow immediate regeneration of the paper figures/tables. Results depend on system size (e.g., 4×2 and 5×2 ladders), disorder type (random vs quasiperiodic), and sampling parameters documented in the notebook. License Choose one of the following (recommended): MIT License (best for code + data reuse), or CC BY 4.0 (best if you want strict attribution for all materials). Citation If you use this package, please cite: The Zenodo DOI for this record (automatically generated by Zenodo), And the associated manuscript/preprint (add arXiv or journal reference here once available).
Standard quantum verification and certification protocols often assume that experimental sources emit independent and identically distributed (i.i.d.) states. In realistic scenarios, however, temporal drift, memory effects, feedback, and correlated noise can violate this assumption, causing standard analyses to underestimate uncertainty and overestimate device performance. Here, we introduce a general framework for quantum verification and certification in which the state produced in each round may depend arbitrarily on the preceding experimental history. Our method provides rigorous confidence intervals for observable expectation values averaged over the experimental trajectory. The framework covers both full verification, in which every state is measured, and spot-checking certification, in which a randomly selected subset is tested to certify the average observable value of the complementary, remaining systems. We apply the framework to importance-sampled Pauli measurements for direct fidelity estimation, energy estimation, and entanglement witnessing, and to classical shadows, allowing multiple observables to be certified simultaneously from the same test data. We illustrate the framework numerically and demonstrate it experimentally through Bell-state certification on a quantum processor.