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Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure

Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter...

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Autores principales: Tranter, Andrew, Love, Peter J., Mintert, Florian, Wiebe, Nathan, Coveney, Peter V.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2019
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7514563/
http://dx.doi.org/10.3390/e21121218
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author Tranter, Andrew
Love, Peter J.
Mintert, Florian
Wiebe, Nathan
Coveney, Peter V.
author_facet Tranter, Andrew
Love, Peter J.
Mintert, Florian
Wiebe, Nathan
Coveney, Peter V.
author_sort Tranter, Andrew
collection PubMed
description Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
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spelling pubmed-75145632020-11-09 Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure Tranter, Andrew Love, Peter J. Mintert, Florian Wiebe, Nathan Coveney, Peter V. Entropy (Basel) Article Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian. MDPI 2019-12-13 /pmc/articles/PMC7514563/ http://dx.doi.org/10.3390/e21121218 Text en © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Tranter, Andrew
Love, Peter J.
Mintert, Florian
Wiebe, Nathan
Coveney, Peter V.
Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure
title Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure
title_full Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure
title_fullStr Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure
title_full_unstemmed Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure
title_short Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure
title_sort ordering of trotterization: impact on errors in quantum simulation of electronic structure
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7514563/
http://dx.doi.org/10.3390/e21121218
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