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

The Journal of Physical Chemistry Letters

Does Your Self-Consistent Field Program Know How Wrong It Is? Introducing Quantum Chemistry Uncertainty Propagation

Anna Diaz Farias, Shalith Chamantha, Jonathan M. Waldrop, Eric Weber, Theresa L. Windus, Ryan M. Richard

Published Sep 22, 2026• 0 citations
Publisher

Abstract

Quantum chemistry relies on numerical approximations. Establishing the correctness and limits of these approximations traditionally requires laborious benchmarking. We introduce a more direct approach, quantum chemistry uncertainty propagation (QCUP), and apply it to a full iterative self-consistent field (SCF) calculation. QCUP-SCF estimates uncertainty from floating-point arithmetic, primitive-Gaussian screening, and incomplete SCF convergence, and propagates it to the final energy and density. Across all systems tested, predictions are typically within 1 kcal/mol compared to analytic error bars and within an order of magnitude of the true error ─ enough to establish significant figures and support runtime decision making. We also show how thresholded-affine forms improve on interval bounds by exploiting guaranteed error cancellation. This is, to our knowledge, the first end-to-end forward uncertainty quantification demonstration for an iterative ab initio method, and the first applied to both energy and wave function.

DOI
10.1021/acs.jpclett.6c02837

Related papers

Accelerating Wave Function Convergence in Interactive Quantum Chemical Reactivity Studies

Adrian H. Mühlbach, Alain C. Vaucher, Markus Reiher

The inherently high computational cost of iterative self-consistent field (SCF) methods proves to be a critical issue delaying visual and haptic feedback in real-time quantum chemistry. In this work, we introduce two schemes for SCF acceleration. They provide a guess for the initial density matrix of the SCF procedure generated by extrapolation techniques. SCF optimizations then converge in fewer iterations, which decreases the execution time of the SCF optimization procedure. To benchmark the proposed propagation schemes, we developed a test bed for performing quantum chemical calculations on sequences of molecular structures mimicking real-time quantum chemical explorations. Explorations of a set of six model reactions employing the semi-empirical methods PM6 and DFTB3 in this testing environment showed that the proposed propagation schemes achieved speedups of up to 30% as a consequence of a reduced number of SCF iterations.

arXiv:1512.02111•Journal of Chemical Theory and Computation•Jan 20, 2016•23 citations

Quantum-Accelerated Self-Consistent Field: A Hybrid Algorithm

Alexis Ralli, Tim Weaving, Thomas M. Bickley +2 more

We present the Grover adaptive search self-consistent field (GAS-SCF) algorithm. GAS-SCF leverages quantum arithmetic to construct an efficient oracle that marks target states (Fock states) which improve upon some initial classical energy estimate. Amplitude amplification then increases the probability of measuring these states. This approach offers a theoretical quadratic speed-up for the optimization problem encountered in SCF quantum chemistry and establishes a baseline against which structured optimization algorithms, such as QAOA and DQI may be compared. In this work, we classically simulate three examples as proofs of concept of the algorithm, the largest consisting of 26 qubits. We then extend our analysis to two larger systems, with O3 representing the largest case at 330 qubits. These examples are chosen to probe classically challenging SCF regimes. Achieving chemically relevant applications of GAS-SCF will require large-scale, fault-tolerant quantum hardware.

quant-pharXiv:2606.20176•Jun 19, 2026•0 citations

Bridging Quantum Chemistry and MaxCut: Classical Performance Guarantees and Quantum Algorithms for the Hartree–Fock Method

Alexis Ralli, Tim Weaving, Peter V. Coveney +1 more

In quantum chemistry, self-consistent field (SCF) algorithms define a nonlinear optimization problem, with both continuous and discrete components. In this work, we derive Hartree–Fock-inspired SCF algorithms that can be exactly written as a sequence of Quadratic Unconstrained Spin/Binary Optimization problems (QUSO/QUBO). We reformulate the optimization problem as a series of MaxCut graph problems, which can be efficiently solved using semidefinite programming techniques. This procedure provides performance guarantees at each SCF step, irrespective of the complexity of the optimization landscape. We numerically demonstrate the QUBO-SCF and MaxCut-SCF methods by studying the hydroxide anion OH – and molecular Nitrogen N 2 . The largest problem addressed in this study involves a system comprised of 220 qubits (equivalently, spin–orbitals). Our results show that QUBO-SCF and MaxCut-SCF suffer much less from internal instabilities compared with conventional SCF calculations. Additionally, we show that the new SCF algorithms can enhance single-reference methods, such as configuration interaction. Finally, we explore how quantum algorithms for optimization can be applied to the QUSO problems arising from the Hartree–Fock method. Four distinct hybrid-quantum classical approaches are introduced: GAS-SCF, QAOA-SCF, QA-SCF and DQI-SCF.

arXiv:2506.04223•Journal of Chemical Theory and Computation•Sep 23, 2025•2 citations

Quantum chemistry using the density matrix renormalization group II

Alexander O. Mitrushenkov, Roberto Linguerri, Paolo Palmieri +1 more

We have compared different strategies for ab initio quantum chemistry density matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the reference complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approximate occupation numbers or a priori, using a Møller–Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping constant the number m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate estimates of the full configuration interaction energy, making the expansion independent on the actual values of the two parameters m and λ. We propose to generate the factors for the two blocks from ground and excited eigenvectors of the Hamiltonian matrix.

The Journal of Chemical Physics•Aug 14, 2003•72 citations

Semiempirical Quantum Chemistry in the Age of ab initio Data and Differentiable Programming: I. Differentiable Molecular Orbital Theory

Martin Stöhr, Todd J. Martı́nez

Semiempirical quantum chemistry (SQC) methods offer fast quantum chemical insights by constructing and solving a parametric effective minimal basis Fock matrix. Establishing suitable parameterizations has long been a challenging and time-consuming task involving tedious grid searches or costly finite-difference gradients of carefully crafted loss functions based on select experimental data. The growing availability of differentiable programming environments that leverage algorithmic differentiation to obtain complex derivatives together with access to a wealth of reliable reference data from ab initio calculations offers a new and more efficient approach. In this work, we extend a previous, basic implementation of SQC methods in PyTorch [J. Chem. Theory Comput. 16, 4951-4962 (2020)] by including global algorithmic considerations into the code design. This allows for improved general applicability and establishes a robust back-end for rapid SQC parameterizations. In particular, we address the general differentiability of the eigensolver and the iterative SCF procedure. The new implementation offers drastic improvements in computing costs as well as memory footprint and provides increased stability in gradient evaluation. We highlight the importance of these advances and their improvements over existing formulations and illustrate their role in the context of SQC parameterization.

Journal of Chemical Theory and Computation•Jan 8, 2026•1 citation

Quantum Chemistry on Graphical Processing Units. 2. Direct Self-Consistent-Field Implementation

Ivan S. Ufimtsev, Todd J. Martı́nez

We demonstrate the use of graphical processing units (GPUs) to carry out complete self-consistent-field calculations for molecules with as many as 453 atoms (2131 basis functions). Speedups ranging from 28× to 650× are achieved as compared to a mature third-party quantum chemistry program (GAMESS) running on a traditional CPU. The computational organization used to construct the Coulomb and exchange operators is discussed. We also present results using three GPUs in parallel, combining coarse and fine-grained parallelism.

Journal of Chemical Theory and Computation•Mar 6, 2009•451 citations
Quantum chemistry relies on numerical approximations. Establishing the correctness and limits of these approximations traditionally requires laborious benchmarking. We introduce a more direct approach, quantum chemistry uncertainty propagation (QCUP), and apply it to a full iterative self-consistent field (SCF) calculation. QCUP-SCF estimates uncertainty from floating-point arithmetic, primitive-Gaussian screening, and incomplete SCF convergence, and propagates it to the final energy and density. Across all systems tested, predictions are typically within 1 kcal/mol compared to analytic error bars and within an order of magnitude of the true error ─ enough to establish significant figures and support runtime decision making. We also show how thresholded-affine forms improve on interval bounds by exploiting guaranteed error cancellation. This is, to our knowledge, the first end-to-end forward uncertainty quantification demonstration for an iterative ab initio method, and the first applied to both energy and wave function.
The inherently high computational cost of iterative self-consistent field (SCF) methods proves to be a critical issue delaying visual and haptic feedback in real-time quantum chemistry. In this work, we introduce two schemes for SCF acceleration. They provide a guess for the initial density matrix of the SCF procedure generated by extrapolation techniques. SCF optimizations then converge in fewer iterations, which decreases the execution time of the SCF optimization procedure. To benchmark the proposed propagation schemes, we developed a test bed for performing quantum chemical calculations on sequences of molecular structures mimicking real-time quantum chemical explorations. Explorations of a set of six model reactions employing the semi-empirical methods PM6 and DFTB3 in this testing environment showed that the proposed propagation schemes achieved speedups of up to 30% as a consequence of a reduced number of SCF iterations.
We present the Grover adaptive search self-consistent field (GAS-SCF) algorithm. GAS-SCF leverages quantum arithmetic to construct an efficient oracle that marks target states (Fock states) which improve upon some initial classical energy estimate. Amplitude amplification then increases the probability of measuring these states. This approach offers a theoretical quadratic speed-up for the optimization problem encountered in SCF quantum chemistry and establishes a baseline against which structured optimization algorithms, such as QAOA and DQI may be compared. In this work, we classically simulate three examples as proofs of concept of the algorithm, the largest consisting of 26 qubits. We then extend our analysis to two larger systems, with O3 representing the largest case at 330 qubits. These examples are chosen to probe classically challenging SCF regimes. Achieving chemically relevant applications of GAS-SCF will require large-scale, fault-tolerant quantum hardware.
In quantum chemistry, self-consistent field (SCF) algorithms define a nonlinear optimization problem, with both continuous and discrete components. In this work, we derive Hartree–Fock-inspired SCF algorithms that can be exactly written as a sequence of Quadratic Unconstrained Spin/Binary Optimization problems (QUSO/QUBO). We reformulate the optimization problem as a series of MaxCut graph problems, which can be efficiently solved using semidefinite programming techniques. This procedure provides performance guarantees at each SCF step, irrespective of the complexity of the optimization landscape. We numerically demonstrate the QUBO-SCF and MaxCut-SCF methods by studying the hydroxide anion OH – and molecular Nitrogen N 2 . The largest problem addressed in this study involves a system comprised of 220 qubits (equivalently, spin–orbitals). Our results show that QUBO-SCF and MaxCut-SCF suffer much less from internal instabilities compared with conventional SCF calculations. Additionally, we show that the new SCF algorithms can enhance single-reference methods, such as configuration interaction. Finally, we explore how quantum algorithms for optimization can be applied to the QUSO problems arising from the Hartree–Fock method. Four distinct hybrid-quantum classical approaches are introduced: GAS-SCF, QAOA-SCF, QA-SCF and DQI-SCF.
We have compared different strategies for ab initio quantum chemistry density matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the reference complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approximate occupation numbers or a priori, using a Møller–Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping constant the number m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate estimates of the full configuration interaction energy, making the expansion independent on the actual values of the two parameters m and λ. We propose to generate the factors for the two blocks from ground and excited eigenvectors of the Hamiltonian matrix.
Semiempirical quantum chemistry (SQC) methods offer fast quantum chemical insights by constructing and solving a parametric effective minimal basis Fock matrix. Establishing suitable parameterizations has long been a challenging and time-consuming task involving tedious grid searches or costly finite-difference gradients of carefully crafted loss functions based on select experimental data. The growing availability of differentiable programming environments that leverage algorithmic differentiation to obtain complex derivatives together with access to a wealth of reliable reference data from ab initio calculations offers a new and more efficient approach. In this work, we extend a previous, basic implementation of SQC methods in PyTorch [J. Chem. Theory Comput. 16, 4951-4962 (2020)] by including global algorithmic considerations into the code design. This allows for improved general applicability and establishes a robust back-end for rapid SQC parameterizations. In particular, we address the general differentiability of the eigensolver and the iterative SCF procedure. The new implementation offers drastic improvements in computing costs as well as memory footprint and provides increased stability in gradient evaluation. We highlight the importance of these advances and their improvements over existing formulations and illustrate their role in the context of SQC parameterization.
We demonstrate the use of graphical processing units (GPUs) to carry out complete self-consistent-field calculations for molecules with as many as 453 atoms (2131 basis functions). Speedups ranging from 28× to 650× are achieved as compared to a mature third-party quantum chemistry program (GAMESS) running on a traditional CPU. The computational organization used to construct the Coulomb and exchange operators is discussed. We also present results using three GPUs in parallel, combining coarse and fine-grained parallelism.