A diagrammatic approach to variational quantum ansatz construction

Variational quantum eigensolvers (VQEs) are a promising class of quantum algorithms for preparing approximate ground states in near-term quantum devices. Minimizing the error in such an approximation requires designing ansatzes using physical considerations that target the studied system. One such consideration is size-extensivity, meaning that the ground state quantum correlations are to be compactly represented in the ansatz. On digital quantum computers, however, the size-extensive ansatzes usually require expansion via Trotter-Suzuki methods. These introduce additional costs and errors to the approximation. In this work, we present a diagrammatic scheme for the digital VQE ansatzes, which is size-extensive but does not rely on Trotterization. We start by designing a family of digital ansatzes that explore the entire Hilbert space with the minimum number of free parameters. We then demonstrate how one may compress an arbitrary digital ansatz, by enforcing symmetry constraints of the target system, or by using them as parent ansatzes for a hierarchy of increasingly long but increasingly accurate sub-ansatzes. We apply a perturbative analysis and develop a diagrammatic formalism that ensures the size-extensivity of generated hierarchies. We test our methods on a short spin chain, finding good convergence to the ground state in the paramagnetic and the ferromagnetic phase of the transverse-field Ising model.

Kernel Charge Equilibration: Efficient and Accurate Prediction of Molecular Dipole Moments with a Machine-Learning Enhanced Electron Density Model

State-of-the-art machine learning (ML) interatomic potentials use local representations of atomic environments to ensure linear scaling and size-extensivity. This implies a neglect of long-range interactions, most prominently related to electrostatics. To overcome this limitation, we herein present a ML framework for predicting charge distributions and their interactions termed kernel Charge Equilibration (kQEq). This model is based on classical charge equilibration models like QEq, expanded with an environment dependent electronegativity. In contrast to previously reported neural network models with a similar concept, kQEq takes advantage of the linearity of both QEq and Kernel Ridge Regression to obtain a closed-form linear algebra expression for training the models. Furthermore, we avoid the ambiguity of charge partitioning schemes by using dipole moments as reference data. As a first application, we show that kQEq can be used to generate accurate and highly data-efficient models for molecular dipole moments.

Size-Extensive Molecular Machine Learning with Global Descriptors

<div> <div> <div> <p>Machine learning (ML) models are increasingly used to predict molecular prop- erties in a high-throughput setting at a much lower computational cost than con- ventional electronic structure calculations. Such ML models require descriptors that encode the molecular structure in a vector. These descriptors are generally designed to respect the symmetries and invariances of the target property. However, size- extensivity is usually not guaranteed for so-called global descriptors. In this contri- bution, we show how extensivity can be build into ML models with global descriptors such as the Many-Body Tensor Representation. Properties of extensive and non- extensive models for the atomization energy are systematically explored by training on small molecules and testing on small, medium and large molecules. Our result shows that the non-extensive model is only useful in the size-range of its training set, whereas the extensive models provide reasonable predictions across large size differences. Remaining sources of error for the extensive models are discussed. </p> </div> </div> </div>

Size-Extensive Molecular Machine Learning with Global Descriptors

<div> <div> <div> <p>Machine learning (ML) models are increasingly used to predict molecular prop- erties in a high-throughput setting at a much lower computational cost than con- ventional electronic structure calculations. Such ML models require descriptors that encode the molecular structure in a vector. These descriptors are generally designed to respect the symmetries and invariances of the target property. However, size- extensivity is usually not guaranteed for so-called global descriptors. In this contri- bution, we show how extensivity can be build into ML models with global descriptors such as the Many-Body Tensor Representation. Properties of extensive and non- extensive models for the atomization energy are systematically explored by training on small molecules and testing on small, medium and large molecules. Our result shows that the non-extensive model is only useful in the size-range of its training set, whereas the extensive models provide reasonable predictions across large size differences. Remaining sources of error for the extensive models are discussed. </p> </div> </div> </div>

The Pole Structure of the Dynamical Polarizability Tensor in Equation-of-Motion Coupled-Cluster Theory

<div>In this Letter, we investigate the pole structure of dynamical polarizabilities computed within the equation-of-motion coupled-cluster (EOM-CC) theory. We show, both theoretically and numerically, that approximate EOM-CC schemes such as, for example, the EOM-CC singles and doubles (EOM-CCSD) model exhibit an incorrect pole structure in which the poles that reflect the excitations from the target state (i.e., the EOM-CC state) are supplemented by artificial poles due to excitations from the coupled-cluster (CC) reference state. These artificial poles can be avoided</div><div>by skipping the amplitude response and reverting to a sum-over-states formulation. While numerical results are generally in favor of such a solution, its major drawback</div><div>is that this scheme violates size extensivity.</div>

The Pole Structure of the Dynamical Polarizability Tensor in Equation-of-Motion Coupled-Cluster Theory

<div>In this Letter, we investigate the pole structure of dynamical polarizabilities computed within the equation-of-motion coupled-cluster (EOM-CC) theory. We show, both theoretically and numerically, that approximate EOM-CC schemes such as, for example, the EOM-CC singles and doubles (EOM-CCSD) model exhibit an incorrect pole structure in which the poles that reflect the excitations from the target state (i.e., the EOM-CC state) are supplemented by artificial poles due to excitations from the coupled-cluster (CC) reference state. These artificial poles can be avoided</div><div>by skipping the amplitude response and reverting to a sum-over-states formulation. While numerical results are generally in favor of such a solution, its major drawback</div><div>is that this scheme violates size extensivity.</div>