Quantum computing methods for electronic states of the water molecule

Abstract

We compare recently proposed methods to compute the electronic state energies of the water molecule on a quantum computer. The methods include the phase estimation algorithm based on Trotter decomposition, the phase estimation algorithm based on the direct implementation of the Hamiltonian, direct measurement based on the implementation of the Hamiltonian and a specific variational quantum eigensolver, Pairwise VQE. After deriving the Hamiltonian using STO-3G basis, we first explain how each method works and then compare the simulation results in terms of gate complexity and the number of measurements for the ground state of the water molecule with different O-H bond lengths. Moreover, we present the analytical analyses of the error and the gate-complexity for each method. While the required number of qubits for each method is almost the same, the number of gates and the error vary a lot. In conclusion, among methods based on the phase estimation algorithm, the second order direct method provides the most efficient circuit implementations in terms of the gate complexity. With large scale quantum computation, the second order direct method seems to be better for large molecule systems. Moreover, Pairewise VQE serves the most practical method for near-term applications on the current available quantum computers. Finally the possibility of extending the calculation to excited states and resonances is discussed.

Quantum-classical hybrid algorithm using an error-mitigating N -representability condition to compute the Mott metal-insulator transition

“…A number of quantum algorithms have been developed which exhibit lower scaling than their classical counterparts [3][4][5], but emerging quantum hardware is far from being capable of long coherence times, arbitrary connectivity and low gate error, which are requirements for most of these algorithms. As a consequence, efforts to maximally utilize the available devices have taken inspiration from quantum and classical regimes alike [6][7][8][9][10]. In particular, hybrid quantum-classical algorithms have been developed, which attempt to separate efficiently quantum and classical components of a problem [11][12][13].…”

Hybrid Quantum-Classical Neural Network for Calculating Ground State Energies of Molecules

“…For H , we use the Jordan–Wigner transformation [ 36 ] to get a 4-qubit Hamiltonian. We decided to apply the complete active space (CAS) approach [ 37 , 38 ], which divides the orbitals into inactive orbitals such as always occupied low energy orbitals and always unoccupied high energy orbitals, and active orbitals, to reduce the number of qubits of LiH and BeH Hamiltonian [ 4 , 11 ] and the reduced Hamiltonian is only of the active orbitals. For LiH, we assume the first two lowest energy spin orbitals are always occupied and use the binary code transformation [ 39 ] considering spin symmetry to save two qubits.…”

“…Quantum computing has shown its great potential in advancing quantum chemistry research [ 1 ]. Many quantum algorithms have been proposed to solve quantum chemistry problems [ 2 , 3 , 4 ], such as the Phase Estimation Algorithm; Aspuru-Guzik et al [ 5 , 6 , 7 , 8 ] to calculate eigenstate energies of simple molecules; the Variational Quantum Eigensolver (VQE) [ 9 , 10 , 11 ] to solve electronic structure problems; quantum algorithms for open quantum dynamics [ 12 ]; and benchmark calculations for two-electron molecules conducted on quantum computers [ 13 ]. Using quantum computing techniques to perform machine learning tasks [ 14 ] has also received much attention recently including quantum data classification [ 15 , 16 ], quantum generative learning [ 17 , 18 ], and quantum neural network approximating nonlinear functions [ 19 ].…”

Context-aware quantum simulation of a matrix stored in quantum memory

“…Variational quantum eigensolver [10,11] is generally applied to the quantum chemistry problems that are represented by the electronic Hamiltonian in the second quantization by transforming the Hamiltonian to the sum of Pauli operators, which are the products of the Pauli spin matrices(e.g. [7,17,18]). Assume the electronic Hamiltonian is H = i h i H i , where H i is a Pauli operator and h i is the corresponding coefficient.…”

“…4 shows the ground state energy curve of H 2 obtained by the simulation based on this modified VQE. In the simulation, the 4-qubit Hamiltonian of H 2 is calculated by openfermion package [20] using STO-3G basis set, and the hardware-efficient ansatz is prepared by 3layer pairwise design in [7].…”