Program Synthesis of Sparse Algorithms for Wave Function andEnergy Prediction in Grid-Based Quantum Simulations

We have recently shown how program synthesis(PS), or the concept ofself-writing code, can generate novelalgorithms that solve the vibrational Schro''dinger equation,providing approximations to the allowed wave functions forbound, one-dimensional (1-D) potential energy surfaces (PESs).The resulting algorithms use a grid-based representation of theunderlying wave function psi(x) and PESV(x), providing codeswhich represent approximations to standard discrete variablerepresentation (DVR) methods. In this Article, we show how thisinductive PS strategy can be improved and modified to enable prediction of both vibrational wave functionsandenergy eigenvaluesof representative model PESs (both 1-D and multidimensional). We show that PS can generate algorithms that offer someimprovements in energy eigenvalue accuracy over standard DVR schemes; however, we also demonstrate that PS can identifyaccurate numerical methods that exhibit desirable computational features, such as employing very sparse (tridiagonal) matrices. Theresulting PS-generated algorithms are initially developed and tested for 1-D vibrational eigenproblems, before solution ofmultidimensional problems is demonstrated; wefind that our new PS-generated algorithms can reduce calculation times for grid-based eigenvector computation by an order of magnitude or more. More generally, with further development and optimization, weanticipate that PS-generated algorithms based on effective Hamiltonian approximations, such as those proposed here, could be usefulin direct simulations of quantum dynamics via wave function propagation and evaluation of molecular electronic structure.

Automated summary

This article demonstrates how program synthesis can be used to generate algorithms for approximating vibrational wave functions and energy levels. By improving and modifying the synthesis strategy, the resulting algorithms show improved accuracy and reduced computation time, making them potentially useful for direct simulations of quantum dynamics.