Electronic Structure Calculations and the Ising Hamiltonian

Abstract

The exact solution of the Schrödinger equation for atoms, molecules and extended systems continues to be a "Holy Grail" problem for the field of atomic and molecular physics since inception. Recently, breakthroughs have been made in the development of hardwareefficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins through an Ising-type Hamiltonian. One of the most vital questions associated with these new devices is: "Can these machines be used to perform electronic structure calculations?" In this study, we discuss the general standard procedure used by these devices and show that there is an exact mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. The simulation results of the transformed Ising Hamiltonian for H2, He2, HeH + , and LiH molecules 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.The determination of solutions to the Schrödinger equation is fundamentally difficult as the dimensionality of the corresponding Hilbert space increases exponentially with the number of particles in the system, requiring a commensurate increase in computational resources. Modern quantum chemistry -faced with difficulties associated with solving the Schrödinger equation to chemical accuracy (∼1 kcal/mole) -has largely become an endeavor to find approximate methods. A few products of this effort from the past few decades include methods such as: ab initio, Density Functional, Density Matrix, Algebraic, Quantum Monte Carlo and Dimensional Scaling [1,2,3,4]. However, all methods hitherto devised face the insurmountable challenge of escalating computational resource requirements as the calculation is extended either to higher accuracy or to larger systems. Computational complexity in electronic structure calculations [5,6,7] suggests that these restrictions are an inherent difficulty associated with simulating quantum systems.Electronic structure algorithms developed for quantum computers provide a new promising route to advance the field of electronic structure calculations for large systems [8,9]. Recently, there has been an attempt at using an adiabatic quantum computing model -as is implemented on the D-Wave machine -to perform electronic structure calculations [10]. The fundamental concept behind the adiabatic quantum computing (AQC) method is to define a problem Hamiltonian, H P , engineered to have its ground state encode the solution of a corresponding computational problem. The system is initialized in the ground state of a beginning Hamiltonian, H B , which is easily solved classically. The system is then allowed to evolve adiabatically as: The largest scale implementation of AQC to date is by D-Wave Systems [11,12]. In the case of the DWave device, the physical process undertaken which acts as an adiabatic evolution is more broadly called quantum annealing (QA). The quantum processor...

Quantum machine learning for electronic structure calculations

“…Equation ( 5 ) is then transformed to Pauli matrices representation, which is achieved by the Jordan-Wigner transformation 34 . The final electronic structure Hamiltonian takes the general form with where σ x , σ y , σ z are Pauli matrices and I is the identity matrix: 35 …”

An adaptive variational algorithm for exact molecular simulations on a quantum computer

“…This was followed by several theoretical studies on VQE [7,[11][12][13][14][15][16][17] and demonstrations on other hardware such as superconducting qubits [7,16,18] and trapped ions [19,20]. Other approaches have been pursued as well, including methods for adiabatic quantum computation [21] and quantum machine learning [22].…”

Solving Quantum Chemistry Problems with a D-Wave Quantum Annealer

“…1. To accomplish this, we use the method proposed by Xia et al [25]. In this section, we shortly recapitulate the method, for a detailed overview, cf.…”

“…By increasing r, we are able to grasp more quantum effects and therefore are able to get a more precise estimation of the true ground state energy. In the following, we use this mapping and the proposed algorithm in [25] to find a classical spin glass approximation of our electronic structure Hamiltonian, which we then solve by using the methods of quantum annealing.…”

“…While it has been successfully proven that such systems can in general be used for solving combinatorial optimization problems, they have not been used to solve chemistry applications. In this paper we apply a mapping, put forward by Xia et al [25], from a quantum chemistry Hamiltonian to an Ising spin glass formulation and find the ground state energy with a quantum annealer. Additionally we investigate the scaling in terms of needed physical qubits on a quantum annealer with limited connectivity.…”