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Journal of Chemical Theory and Computation

Determining the Numerical Stability of Quantum Chemistry Algorithms

Gerald Knizia, Wenbin Li, Sven Simon, Hans‐Joachim Werner

Published Jul 14, 2011• 29 citations
Publisher

Abstract

We present a simple, broadly applicable method for determining the numerical properties of quantum chemistry algorithms. The method deliberately introduces random numerical noise into computations, which is of the same order of magnitude as the floating point precision. Accordingly, repeated runs of an algorithm give slightly different results, which can be analyzed statistically to obtain precise estimates of its numerical stability. This noise is produced by automatic code injection into regular compiler output, so that no substantial programming effort is required, only a recompilation of the affected program sections. The method is applied to investigate: (i) the numerical stability of the three-center Obara-Saika integral evaluation scheme for high angular momenta, (ii) if coupled cluster perturbative triples can be evaluated with single precision arithmetic, (iii) how to implement the density fitting approximation in Møller-Plesset perturbation theory (MP2) most accurately, and (iv) which parts of density fitted MP2 can be safely evaluated with single precision arithmetic. In the integral case, we find a numerical instability in an equation that is used in almost all integral programs. Due to the results of (ii) and (iv), we conjecture that single precision arithmetic can be applied whenever a calculation is done in an orthogonal basis set and excessively long linear sums are avoided.

DOI
10.1021/ct200239p

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We present a simple, broadly applicable method for determining the numerical properties of quantum chemistry algorithms. The method deliberately introduces random numerical noise into computations, which is of the same order of magnitude as the floating point precision. Accordingly, repeated runs of an algorithm give slightly different results, which can be analyzed statistically to obtain precise estimates of its numerical stability. This noise is produced by automatic code injection into regular compiler output, so that no substantial programming effort is required, only a recompilation of the affected program sections. The method is applied to investigate: (i) the numerical stability of the three-center Obara-Saika integral evaluation scheme for high angular momenta, (ii) if coupled cluster perturbative triples can be evaluated with single precision arithmetic, (iii) how to implement the density fitting approximation in Møller-Plesset perturbation theory (MP2) most accurately, and (iv) which parts of density fitted MP2 can be safely evaluated with single precision arithmetic. In the integral case, we find a numerical instability in an equation that is used in almost all integral programs. Due to the results of (ii) and (iv), we conjecture that single precision arithmetic can be applied whenever a calculation is done in an orthogonal basis set and excessively long linear sums are avoided.
In the present work, we show the feasibility of using single precision in quantum chemistry, especially regarding the computation of electron correlation energy. On the example of the MP2 method, we clearly demonstrate that single precision arithmetic is sufficient for evaluating the molecular two-electron integrals by the use of the Cholesky decomposition method. The evaluation of integrals with single precision arithmetic introduces a negligible error into the computed MP2 correlation energy. In particular, the corresponding error in the MP2 correlation energy amounts to only 10(-7)Eh for the 113-atom taxol molecule in double-valence basis set (1099 basis functions). The practical relevance of our result is that 50% performance gain and 50% reduction in memory demands can be achieved by only minor changes in the existing codes. Our finding opens intriguing perspectives for doing accurate correlated quantum chemistry on specialized floating-point mathematical coprocessors.
Adaptive Algorithms / Convergence Analysis / Best N-term Approximation / Coupled Cluster Theory / Error Estimators In this article, we summarize the results of our numerical analysis for various adaptive schemes in electronic structure calculations. Starting from a general form of iterative schemes used in quantum chemistry, we discuss miscellaneous features which are of importance when passing to an adaptive version of an algorithm in order to provide rigorous estimates. The key features are maintainance of convergence under small perturbations while keeping control of the computational complexity. As a prototype for correlation methods, we consider the configuration interaction (CI) method and discuss a quasi-optimal algorithm for computation of the ground state energy. Our analysis of the coupled cluster method shows that various properties of the CI method can be transferred hereunto as well. First numerical examples demonstrate the potential of the coupled cluster residual, measured in a norm on the dual space, as an error estimator. These studies are supplemented by our result for the best N-term approximation of electron-pair correlations by wavelets. For Hartree-Fock and Kohn-Sham methods, we discuss our results of the convergence analysis for a direct minimization scheme, the best N-term approximation of orbitals and s * -compressibility of the Fock operator which altogether may serve as a basis for an adaptive version of these methods.
Measuring expectation values of observables is an essential ingredient in variational quantum algorithms. A practical obstacle is the necessity of a large number of measurements for statistical convergence to meet requirements of precision, such as chemical accuracy in the application to quantum chemistry computations. Here we propose an algorithm to estimate the expectation value based on its approximate expression as a weighted sum of classically tractable matrix elements with some modulation, where the weight and modulation factors are evaluated by sampling appropriately prepared quantum states in the computational basis on quantum computers. Each of those states is prepared by applying a unitary transformation consisting of at most NNN CNOT gates, where NNN is the number of qubits, to a target quantum state whose expectation value is evaluated. Our algorithm is expected to require fewer measurements than conventional methods for a required statistical precision of the expectation value when the target quantum state is concentrated in particular computational basis states. We provide numerical comparisons of our method with existing ones for measuring electronic ground-state energies (expectation values of electronic Hamiltonians for the lowest-energy states) of various small molecules. Numerical results show that our method can reduce the numbers of measurements to obtain the ground-state energies for a targeted precision by several orders of magnitudes for molecules whose ground states are concentrated. Our results provide another route to measure expectation values of observables, which could accelerate the variational quantum algorithms.
We present QENEX LAB, an open-architecture quantum chemistry platform implementing Hartree-Fock (RHF/UHF), Møller-Plesset perturbation theory (MP2), coupled cluster singles and doubles (CCSD), the perturbative triples correction CCSD(T), unrestricted CCSD (UCCSD), equation-of-motion CCSD (EOM-CCSD), complete active space self-consistent field (CASSCF), and Kohn-Sham density functional theory (B3LYP, LDA) with linear-response time-dependent DFT (LR-TDDFT). Molecular integrals are evaluated via Obara-Saika recurrence relations with optional libcint acceleration. Benchmark calculations on He, H2, H2O, and other small molecules demonstrate sub-nanohartree agreement with PySCF reference values when using the libcint integral path, and sub-millihartree agreement with the native pure-Python integral engine. We present accuracy benchmarks across STO-3G, cc-pVDZ, and cc-pVTZ basis sets for nine molecules, validate mathematical invariants including Brillouin's theorem, density matrix idempotency, and size consistency, and describe supplementary capabilities including molecular property prediction (ADMET, toxicity, pharmacokinetics), inverse molecular design via genetic algorithms, and a SHA-256 Merkle-tree provenance chain for computational reproducibility. The platform comprises 81 Python source files (90,103 lines) operating without external quantum chemistry dependencies in the computation path.
Quantum computation has made considerable progress in the last decade with multiple emerging technologies providing proof-of-principle experimental demonstrations of such calculations. However, these experimental demonstrations of quantum computation face technical challenges due to the noise and errors that arise from imperfect implementation of the technology. Here, we frame the concepts of computational accuracy, result reproducibility, device reliability and program stability in the context of quantum computation. We provide intuitive definitions for these concepts in the context of quantum computation that lead to operationally meaningful bounds on program output. Our assessment highlights the continuing need for statistical analyses of quantum computing program to increase our confidence in the burgeoning field of quantum information science.
The opportunities afforded by near-term quantum computers to calculate the ground-state properties of small molecules depend on the structure of the computational ansatz as well as the errors induced by device noise. Here we investigate the behavior of these noisy quantum circuits using numerical simulations to estimate the accuracy and fidelity of the prepared quantum states relative to the ground truth obtained by conventional means. We implement several different types of ansatz circuits derived from unitary coupled cluster theory for the purposes of estimating the ground-state energy of sodium hydride using the variational quantum eigensolver algorithm. We show how relative error in the energy and the fidelity scale with the levels of gate-based noise, the internuclear configuration, the ansatz circuit depth, and the parameter optimization methods.