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Journal of Physics Conference Series

Automated symbolic algebra for quantum chemistry

So Hirata

Published Sep 1, 2006• 8 citations
Publisher

Abstract

Today, state-of-the-art computational methods and programs for quantum theory for electron correlation may not be developed entirely manually. They are often constructed with the essential aid of symbolic algebra systems automating lengthy and error-prone mathematical derivation and computer implementation, inevitable in these developments. Recent progress in this new paradigm of chemical theory development—complete automation—is reviewed.

DOI
10.1088/1742-6596/46/1/036

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Today, state-of-the-art computational methods and programs for quantum theory for electron correlation may not be developed entirely manually. They are often constructed with the essential aid of symbolic algebra systems automating lengthy and error-prone mathematical derivation and computer implementation, inevitable in these developments. Recent progress in this new paradigm of chemical theory development—complete automation—is reviewed.
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Abstract The rapid and successful strides in quantum chemistry in the past decades can be largely credited to a conspicuous synergy between theoretical and computational advancements. However, the architectural computer archetype that enabled such a progress is approaching a state of more stagnant development. One of the most promising technological avenues for the continuing progress of quantum chemistry is the emerging quantum computing paradigm. This revolutionary proposal comes with several challenges, which span a wide array of disciplines. In chemistry, it implies, among other things, a need to reformulate some of its long established cornerstones in order to adjust to the operational demands and constraints of quantum computers. Due to its relatively recent emergence, much of quantum computing may still seem fairly nebulous and largely unknown to most chemists. It is in this context that here we review and illustrate the basic aspects of quantum information and their relation to quantum computing insofar as enabling simulations of quantum chemistry. We consider some of the most relevant developments in light of these aspects and discuss the current landscape when of relevance to quantum chemical simulations in quantum computers.
In this article, we demonstrate the first-principles computation of quantum chemistry through symbolic computation, using computational algebraic geometry. We have developed a Python program that generates symbolic formulas of one- and two-electron integrals. The approximations of those integrals by multivariate polynomials yield the set of equations required by quantum chemistry. We solve these equations in hybrid ways where numeric and symbolic computations are intertwined. We discuss the possibility of the extension of our scheme to treat the models beyond the Born-Oppenheimer approximation.
Precise prediction and control of chemical reactions and material propertiesthe central goal of precision chemistrytruly rely on accurate and efficient theoretical modeling of chemical systems. In recent years, significant progress has been made in theoretical and computational chemistry. Especially, interdisciplinary studies in the direction of the two emerging computational frontiers, artificial intelligence and quantum computation, are revolutionizing this field. Expressive neural networks can be used to represent many-electron wave functions with high accuracy, capturing strong correlations with a favorable computational cost compared to other methods. Leveraging quantum computation in quantum chemistry is also expected to deliver a quantum advantage in problems including electronic structure calculations. With artificial intelligence and quantum computation, we expect to establish a unified, high-precision, and scalable computational foundation for generating accurate quantum chemical data essential for precision chemistry.
The continuous demand for higher accuracy in computational chemistry necessitates the development of advanced many-body electronic structure methods. However, the derivation and efficient implementation of these theories constitute a significant bottleneck. As the rank of the associated tensors increases, the governing equations explode in complexity, rendering manual implementation labor-intensive, error-prone, and difficult to optimize for modern hardware. To address this challenge, this dissertation presents SeQuant, a comprehensive framework for the automated derivation and parallel implementation of many-body quantum chemistry methods. Built upon a robust symbolic algebra engine, SeQuant allows for the expression of theories in the natural language of second quantization. It automates the transformation of high-level theoretical ansatzes into explicit tensor contraction expressions and subsequently generates optimized, high-performance C++ code. A central innovation of this work is the extension of automated implementation to reduced-scaling methods, which exploit the sparsity inherent in electronic correlation. We introduce a novel "tensor-of-tensors" data structure designed to manage the irregular sparsity patterns of Pair Natural Orbital (PNO) formulations. This development enables the first fully automated implementation of PNO-Coupled Cluster (PNO-CC) methods, bridging the gap between symbolic abstraction and the runtime requirements of sparse tensor algebra. The results demonstrate that SeQuant not only reproduces established dense methods (such as CCSD and CCSDT) with high fidelity but also effectively handles the complexity of sparse, local correlation approaches. By decoupling the complexity of the physics from the details of the implementation, this framework establishes a new paradigm for method development, dramatically accelerating the translation of theoretical insights into computational reality.