Considering recent advancements and successes in the development of efficient quantum algorithms for electronic structure calculations—alongside impressive results using machine learning techniques for computation—hybridizing quantum computing with machine learning for the intent of performing electronic structure calculations is a natural progression. Here we report a hybrid quantum algorithm employing a restricted Boltzmann machine to obtain accurate molecular potential energy surfaces. By exploiting a quantum algorithm to help optimize the underlying objective function, we obtained an efficient procedure for the calculation of the electronic ground state energy for a small molecule system. Our approach achieves high accuracy for the ground state energy for H2, LiH, H2O at a specific location on its potential energy surface with a finite basis set. With the future availability of larger-scale quantum computers, quantum machine learning techniques are set to become powerful tools to obtain accurate values for electronic structures.

Designing quantum algorithms for simulating quantum systems has seen enormous progress, yet few studies have been done to develop quantum algorithms for open quantum dynamics despite its importance in modeling the system-environment interaction found in most realistic physical models. In this work we propose and demonstrate a general quantum algorithm to evolve open quantum dynamics on quantum computing devices. The Kraus operators governing the time evolution can be converted into unitary matrices with minimal dilation guaranteed by the Sz.-Nagy theorem. This allows the evolution of the initial state through unitary quantum gates, while using significantly less resource than required by the conventional Stinespring dilation. We demonstrate the algorithm on an amplitude damping channel using the IBM Qiskit quantum simulator and the IBM Q 5 Tenerife quantum device. The proposed algorithm does not require particular models of dynamics or decomposition of the quantum channel, and thus can be easily generalized to other open quantum dynamical models.

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...

Variational quantum eigensolver (VQE) for electronic structure calculations is believed to be one major potential application of near term quantum computing. Among all proposed VQE algorithms, the unitary coupled cluster singles and doubles excitations (UCCSD) VQE ansatz has achieved high accuracy and received a lot of research interest. However, the UCCSD VQE based on fermionic excitations needs extra terms for the parity when using Jordan–Wigner transformation. Here we introduce a new VQE ansatz based on the particle preserving exchange gate to achieve qubit excitations. The proposed VQE ansatz has gate complexity up-bounded to O(n 4) for all-to-all connectivity where n is the number of qubits of the Hamiltonian. Numerical results of simple molecular systems such as BeH2, H2O, N2, H4 and H6 using the proposed VQE ansatz gives very accurate results within errors about 10−3 Hartree.

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.