Solving the Schrödinger equation using program synthesis

• Published in: The Journal of chemical physics Vol. 155; no. 15; pp. 154102 - 154117
• Main Author: Habershon, Scott
• Format: Journal Article
• Language: English
• Published: Melville American Institute of Physics 21.10.2021
• Subjects: Algorithms; Mathematical analysis; Mathematical functions; Optimization; Polynomials; Potential energy; Quantum chemistry; Schrodinger equation; Sequences; Simulated annealing; Synthesis; Vectors (mathematics); Wave functions
• ISSN: 0021-9606; 1089-7690; 1520-9032; 1089-7690
• DOI: 10.1063/5.0062497

We demonstrate that a program synthesis approach based on a linear code representation can be used to generate algorithms that approximate the ground-state solutions of one-dimensional time-independent Schrödinger equations constructed with bound polynomial potential energy surfaces (PESs). Here, an algorithm is constructed as a linear series of instructions operating on a set of input vectors, matrices, and constants that define the problem characteristics, such as the PES. Discrete optimization is performed using simulated annealing in order to identify sequences of code-lines, operating on the program inputs that can reproduce the expected ground-state wavefunctions ψ(x) for a set of target PESs. The outcome of this optimization is not simply a mathematical function approximating ψ(x) but is, instead, a complete algorithm that converts the input vectors describing the system into a ground-state solution of the Schrödinger equation. These initial results point the way toward an alternative route for developing novel algorithms for quantum chemistry applications.