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Open AccessArticle

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

1
Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA
2
Department of Physics, Imperial College London, London SW7 2AZ, UK
3
Department of Physics, University of Washington, Seattle, WA 98105, USA
4
Pacific Northwest National Laboratory, Richland, WA 98382, USA
5
Centre for Computational Science, University College London, London WC1H 0AJ, UK
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(12), 1218; https://doi.org/10.3390/e21121218
Received: 8 November 2019 / Revised: 3 December 2019 / Accepted: 9 December 2019 / Published: 13 December 2019
(This article belongs to the Special Issue Quantum Information: Fragility and the Challenges of Fault Tolerance)
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. View Full-Text
Keywords: quantum simulation; electronic structure theory; quantum computing quantum simulation; electronic structure theory; quantum computing
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Tranter, A.; Love, P.J.; Mintert, F.; Wiebe, N.; Coveney, P.V. Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure. Entropy 2019, 21, 1218.

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