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

Abstract

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 Trottermore » 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.« less

Authors:

Tranter, Andrew ^[1]; Love, Peter J. ^[1]; Mintert, Florian ^[2]; Wiebe, Nathan ^[3]; Coveney, Peter V. ^[4]

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+ Show Author Affiliations

1. Tufts Univ., Medford, MA (United States). Dept. of Physics and Astronomy
2. Imperial College, London (United Kingdom). Dept. of Physics
3. Univ. of Washington, Seattle, WA (United States). Dept. of Physics; Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
4. Univ. College London (United Kingdom). Centre for Computational Science

Publication Date:

Fri Dec 13 00:00:00 EST 2019

Research Org.:

Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Sponsoring Org.:

USDOE Office of Science (SC); Engineering and Physical Sciences Research Council (EPSRC); National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR)

OSTI Identifier:

1815699

Grant/Contract Number:  

AC05-76RL01830; EP/G037043/1; EP/L00030X/1; PHY-1818914; FA8702-15-D-0001

Resource Type:

Accepted Manuscript

Journal Name:

Entropy

Additional Journal Information:

Journal Volume: 21; Journal Issue: 12; Journal ID: ISSN 1099-4300

Publisher:

MDPI

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; quantum simulation; electronic structure theory; quantum computing