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arXiv:1706.00271 · The Journal of Physical Chemistry B

Electronic Structure Calculations and the Ising Hamiltonian

Rongxin Xia, Teng Bian, Sabre Kais

Published Nov 3, 2017• 98 citations
arXivHTMLPDF

Abstract

Obtaining exact solutions to the Schrödinger equation for atoms, molecules, and extended systems continues to be a “Holy Grail” problem which the fields of theoretical chemistry and physics have been striving to solve since inception. Recent breakthroughs have been made in the development of hardware-efficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins with an Ising-type Hamiltonian. One of the most vital questions pertaining to these new devices is, “Can these machines be used to perform electronic structure calculations?” Within this work, we review the general procedure used by these devices and prove that there is an exact mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. Additionally, we provide simulation results of the transformed Ising Hamiltonian for H 2, He 2, HeH +, and LiH molecules, which match the exact numerical calculations. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware. This is an early step in developing generalized methods on such devices for chemical physics.

DOI
10.1021/acs.jpcb.7b10371

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Obtaining exact solutions to the Schrödinger equation for atoms, molecules, and extended systems continues to be a “Holy Grail” problem which the fields of theoretical chemistry and physics have been striving to solve since inception. Recent breakthroughs have been made in the development of hardware-efficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins with an Ising-type Hamiltonian. One of the most vital questions pertaining to these new devices is, “Can these machines be used to perform electronic structure calculations?” Within this work, we review the general procedure used by these devices and prove that there is an exact mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. Additionally, we provide simulation results of the transformed Ising Hamiltonian for H 2, He 2, HeH +, and LiH molecules, which match the exact numerical calculations. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware. This is an early step in developing generalized methods on such devices for chemical physics.
The exact solution of Schrodinger equation for atoms, molecules and extended systems continues to be a "Holy Grail" problem that the entire field has been striving to solve since its inception. Recently, breakthroughs have been made in the development of quantum annealing and coherent Ising machines capable of simulating hundreds of connected spins interacting with an Ising type Hamiltonian. One of the most vital questions pertaining to these new devices is: 'can these machine be used to perform electronic structure calculations?' Here we discuss the general procedure used by these devices and show that there is an exact mapping between the electronic structure Hamiltonian of the Hydrogen molecule and the Ising Hamiltonian
Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, obtains the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule of how a chemical Hamiltonian can be simulated using a quantum computer.
Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, obtains the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule of how a chemical Hamiltonian can be simulated using a quantum computer.
The accurate computational determination of chemical, materials, biological, and atmospheric properties has a critical impact on a wide range of health and environmental problems, but is deeply limited by the computational scaling of quantum mechanical methods. The complexity of quantum chemical studies arises from the steep algebraic scaling of electron correlation methods and the exponential scaling in studying nuclear dynamics and molecular flexibility. To date, efforts to apply quantum hardware to such quantum chemistry problems have focused primarily on electron correlation. Here, we provide a framework that allows for the solution of quantum chemical nuclear dynamics by mapping these to quantum spin-lattice simulators. Using the example case of a short-strong hydrogen-bonded system, we construct the Hamiltonian for the nuclear degrees of freedom on a single Born-Oppenheimer surface and show how it can be transformed to a generalized Ising model Hamiltonian. We then demonstrate a method to determine the local fields and spin-spin couplings needed to identically match the molecular and spin-lattice Hamiltonians. We describe a protocol to determine the on-site and intersite coupling parameters of this Ising Hamiltonian from the Born-Oppenheimer potential and nuclear kinetic energy operator. Our approach represents a paradigm shift in the methods used to study quantum nuclear dynamics, opening the possibility to solve both electronic structure and nuclear dynamics problems using quantum computing systems.
Abstract Digital quantum computers provide a computational framework for solving the Schrödinger equation for a variety of many‐particle systems. Quantum computing algorithms for the quantum simulation of these systems have recently witnessed remarkable growth, notwithstanding the limitations of existing quantum hardware, especially as a tool for electronic structure computations in molecules. In this review, we provide a self‐contained introduction to emerging algorithms for the simulation of Hamiltonian dynamics and eigenstates, with emphasis on their applications to the electronic structure in molecular systems. Theoretical foundations and implementation details of the method are discussed, and their strengths, limitations, and recent advances are presented. This article is categorized under: Quantum Computing > Algorithms Electronic Structure Theory > Ab Initio Electronic Structure Methods Quantum Computing > Theory Development
The accurate computational determination of chemical, materials, biological, and atmospheric properties has critical impact on a wide range of health and environmental problems, but is deeply limited by the computational scaling of quantum-mechanical methods. The complexity of quantum-chemical studies arises from the steep algebraic scaling of electron correlation methods, and the exponential scaling in studying nuclear dynamics and molecular flexibility. To date, efforts to apply quantum hardware to such quantum chemistry problems have focused primarily on electron correlation. Here, we provide a framework which allows for the solution of quantum chemical nuclear dynamics by mapping these to quantum spin-lattice simulators. Using the example case of a short-strong hydrogen bonded system, we construct the Hamiltonian for the nuclear degrees of freedom on a single Born-Oppenheimer surface and show how it can be transformed to a generalized Ising model Hamiltonian. We then demonstrate a method to determine the local fields and spin-spin couplings needed to identically match the molecular and spin-lattice Hamiltonians. We describe a protocol to determine the on-site and inter-site coupling parameters of this Ising Hamiltonian from the Born-Oppenheimer potential and nuclear kinetic energy operator. Our approach represents a paradigm shift in the methods used to study quantum nuclear dynamics, opening the possibility to solve both electronic structure and nuclear dynamics problems using quantum computing systems.