What can machine learning and the Hellmann-Feynman Theorem teach us about the limits of electron correlation?.
Abstract not provided.
Rackers, Joshua↗
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Ab initio molecular dynamics on quantum computers
Ab initio molecular dynamics (AIMD) is a valuable technique for studying molecules and materials at finite temperatures where the nuclei evolve on potential energy surfaces obtained from accurate electronic structure calculations. In this work, we present an approach to running AIMD simulations on noisy intermediate-scale quantum (NISQ)-era quantum computers. The electronic energies are calculated on a quantum computer using the variational quantum eigensolver (VQE) method. Algorithms for computation of analytical gradients entirely on a quantum computer require quantum fault-tolerant hardware, which is beyond NISQ-era. Therefore, we compute the energy gradients numerically using finite differences, the Hellmann-Feynman theorem, and a correlated sampling technique. This method only requires additional classical calculations of electron integrals for each degree of freedom without any additional computations on a quantum computer beyond the initial VQE run. As a proof of concept, AIMD simulations are demonstrated for the H-2 molecule on IBM quantum devices. In addition, we demonstrate the validity of the method for larger molecules using full configuration interaction wave functions. As quantum hardware and noise mitigation techniques continue to improve, the method can be utilized for studying larger molecular systems.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green’s function methods: Theory and application to GW and GF2
One-particle Green’s function methods can model molecular and solid spectra at zero or non-zero temperatures. One-particle Green’s functions directly provide electronic energies and one-particle properties, such as dipole moment. However, the evaluation of two-particle properties, such as $\langle$S 2 $\rangle$ and $\langle$N 2 $\rangle$, can be challenging because they require a solution of the computationally expensive Bethe–Salpeter equation to find two-particle Green’s functions. We demonstrate that the solution of the Bethe–Salpeter equation can be completely avoided. Applying the thermodynamic Hellmann–Feynman theorem to self-consistent one-particle Green’s function methods, we derive expressions for two-particle density matrices in a general case and provide explicit expressions for GF2 and GW methods. Such density matrices can be decomposed into an antisymmetrized product of correlated one-electron density matrices and the two-particle electronic cumulant of the density matrix. Cumulant expressions reveal a deviation from ensemble representability for GW, explaining its known deficiencies. We analyze the temperature dependence of $\langle$S 2 $\rangle$ and $\langle$N 2 $\rangle$ for a set of small closed-shell systems. Interestingly, both GF2 and GW show a non-zero spin contamination and a non-zero fluctuation of the number of particles for closed-shell systems at the zero-temperature limit.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Demonstration of the rodeo algorithm on a quantum computer
The rodeo algorithm is an efficient algorithm for eigenstate preparation and eigenvalue estimation for any observable on a quantum computer. This makes it a promising tool for studying the spectrum and structure of atomic nuclei as well as other fields of quantum many-body physics. The only requirement is that the initial state has sufficient overlap probability with the desired eigenstate. While it is exponentially faster than well-known algorithms such as phase estimation and adiabatic evolution for eigenstate preparation, it has yet to be implemented on an actual quantum device. In this work, we apply the rodeo algorithm to determine the energy levels of a random one-qubit Hamiltonian, resulting in a relative error of 0.08% using mid-circuit measurements on the IBM Q device Casablanca. This surpasses the accuracy of directly-prepared eigenvector expectation values using the same quantum device. We take advantage of the high-accuracy energy determination and use the Hellmann-Feynman theorem to compute eigenvector expectation values for a different random one-qubit observable. For the Hellmann-Feynman calculations, we find a relative error of 0.7%. Here, we conclude by discussing possible future applications of the rodeo algorithm for multi-qubit Hamiltonians.
algorithm↗
Computation of forces and stresses in solids: Towards accurate structural optimization with auxiliary-field quantum Monte Carlo
The accurate computation of forces and other energy derivatives has been a long-standing challenge for quantum Monte Carlo methods. A number of technical obstacles contribute to this challenge. We discuss how these obstacles can be removed with the auxiliary-field quantum Monte Carlo (AFQMC) approach. AFQMC is a general, high-accuracy, many-body total-energy method for molecules and solids. The implementation of back-propagation for pure estimators allows direct calculation of gradients of the energy via the Hellmann-Feynman theorem. A planewave basis with norm-conserving pseudopotentials is used for the study of periodic bulk materials. Completeness of the planewave basis minimizes the effect of so-called Pulay terms. The ionic pseudopotentials, which can be incorporated in AFQMC in exactly the same manner as in standard independent-electron methods, regulate the force and stress estimators and eliminate any potential divergence of the Monte Carlo variances. The resulting approach allows applications of full geometry optimizations in bulk materials. As a result, it also paves the way for many-body computations of the phonon spectrum in solids.
36 MATERIALS SCIENCE↗
Accurate Hellmann–Feynman forces from density functional calculations with augmented Gaussian basis sets
The Hellmann–Feynman (HF) theorem provides a way to compute forces directly from the electron density, enabling efficient force calculations for large systems through machine learning (ML) models for the electron density. The main issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upward of 10 eV/Å in forces. In this work, we demonstrate that if a suitably augmented Gaussian basis set is used for density functional calculations, the Pulay force can be suppressed, and HF forces can be computed as accurately as analytical forces with state-of-the-art basis sets, allowing geometry optimization and molecular dynamics to be reliably performed with HF forces. Our results pave a clear path forward for the accurate and efficient simulation of large systems using ML densities and the HF theorem.
Pathak, Shivesh (ORCID:0000000172798195)↗