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What can machine learning and the Hellmann-Feynman Theorem teach us about the limits of electron correlation?.
Abstract not provided.
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.
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.
Dual-cone variational calculation of the two-electron reduced density matrix
The computation of strongly correlated quantum systems is challenging because of its potentially exponential scaling in the number of electron configurations. Variational calculation of the two-electron reduced density matrix (2-RDM) without the many-electron wave function exploits the pairwise nature of the electronic Coulomb interaction to compute a lower bound on the ground-state energy with polynomial computational scaling. Recently, a dual-cone formulation of the variational 2-RDM calculation was shown to generate the ground-state energy, albeit not the 2-RDM, at a substantially reduced computational cost, especially for higher $\textit{N}$-representability conditions such as the T2 constraint. Here we generalize the dual-cone variational 2-RDM method to compute not only the ground-state energy but also the 2-RDM. The central result is that we can compute the 2-RDM from a generalization of the Hellmann-Feynman theorem. Specifically, we prove that in the Lagrangian formulation of the dual-cone optimization the 2-RDM is the Lagrange multiplier. Further, we apply the method to computing the energies and properties of strongly correlated electrons—including atomic charges, electron densities, dipole moments, and orbital occupations—in an illustrative hydrogen chain and the nitrogen-fixation catalyst FeMoco. The dual variational computation of the 2-RDM with T2 or higher N -representability conditions provides a polynomially scaling approach to strongly correlated molecules and materials with significant applications in atomic and molecular and condensed-matter chemistry and physics.
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.