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 nd 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, nding an optimal strategy for ordering Trotter sequences is di cult. 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 nd that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than kcal/mol.Relative to this, the di ference 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.

Measurement reduction in variational quantum algorithms

“…In order to assess whether this strategy is viable for molecular Hamiltonians, we generated coloring schemes for 65 Hamiltonians (previously used in Refs. [59,60] and described in Appendix C). Geometry specifications were obtained from the NIST CCBDB database [61].…”

Efficient Quantum Simulation of an Anti-P-Pseudo-Hermitian Two-Level System

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“…Enlightened by Feynman’s thinking that simulates physics using nature [ 1 ], plenty of scientists manage to turn it into reality. As near-term quantum computers are available in some ways [ 2 , 3 ], quantum simulation has opened an effective and efficient way to investigate novel systems and phenomena related to both Hermitian [ 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 ] and non-Hermitian (NH) [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ] Hamiltonians. Especially for the later one, quantum simulation has become the main method for experimental investigations in quantum level, which a non-Hermitian system can be constructed, operated, and observed in a subspace of a controllable quantum system.…”

A quantum computing view on unitary coupled cluster theory

“…The same problem arises, whenever an exponential of non-commuting operators is to be Trotterised, such as in Hamiltonian simulation e −iĤ t , whenever individual terms of the Hamiltonian do not commute. 81…”