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 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 for 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

Understanding Quantum Technologies 2022

Understanding Quantum Technologies 2022 is a creative-commons ebook that provides a unique 360 degrees overview of quantum technologies from science and technology to geopolitical and societal issues. It covers quantum physics history, quantum physics 101, gate-based quantum computing, quantum computing engineering (including quantum error corrections and quantum computing energetics), quantum computing hardware (all qubit types, including quantum annealing and quantum simulation paradigms, history, science, research, implementation and vendors), quantum enabling technologies (cryogenics, control electronics, photonics, components fabs, raw materials), quantum computing algorithms, software development tools and use cases, unconventional computing (potential alternatives to quantum and classical computing), quantum telecommunications and cryptography, quantum sensing, quantum technologies around the world, quantum technologies societal impact and even quantum fake sciences. The main audience are computer science engineers, developers and IT specialists as well as quantum scientists and students who want to acquire a global view of how quantum technologies work, and particularly quantum computing. This version is an extensive update to the 2021 edition published in October 2021.Comment: 1132 pages, 920 figures, Letter forma

Quantum chemistry and quantum computers -- Testing the Bravyi-Kitaev mapping and Trotter order optimisations

The quantum simulation of quantum chemistry is often cited as one of the most important potential uses for emerging quantum technology. Quantum computing methods are expected to be able to perform full configuration interaction level quantum chemistry calculations with tractable computational resource requirements. However, early quantum computers are likely to be constrained in both the number of qubits available, and the number of gates that can be performed. Implementing chemical algorithms on such quantum devices will require algorithmic improvements. In this thesis, two avenues for optimisation of the canonical method for the simulation of quantum chemistry on quantum computers are discussed. The Bravyi-Kitaev transformation is an alternative to the Jordan-Wigner transformation, the canonical method of mapping electronic states and operators to states and operators of qubits. The resource implications of this transformation scheme are assessed using a variety of chemical examples. Known techniques for optimising quantum circuits for quantum chemistry are applied, and the implications for the use of the Bravyi-Kitaev transformation are considered. Trotterization – the use of Trotter-Suzuki formulae to approximate the evolution operator for the molecular Hamiltonian – is discussed. In particular, ordering schemes designed to minimise the Trotter error are described. Finally, the importance of Trotter ordering when using the Bravyi-Kitaev transformation is considered. The performance of either transformation scheme is found to be highly dependent on the Trotter ordering and level of optimisation chosen. As such, the relative merits of either transformation scheme are found to be not as simple as previously thought, although in most cases the Bravyi-Kitaev transformation is found to be advantageous. The relative merits of both mapping and ordering schemes are considered in the context of performing quantum simulation of quantum chemistry on real quantum devices. Particular focus is paid to molecular systems which are intractable to classical computational simulation. The results emphasise the need for thorough analysis of optimisation schemes, such that quantum chemistry may indeed be a principal use of early quantum computation.Open Acces