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arXiv:2008.02946 · Journal of Chemical Theory and Computation

Simulating Periodic Systems on a Quantum Computer Using Molecular Orbitals

Jie Liu, Lingyun Wan, Zhenyu Li, Jinlong Yang

Published Oct 19, 2020• 52 citations
arXivHTMLPDF

Abstract

The variational quantum eigensolver (VQE) is one of the most appealing quantum algorithms to simulate electronic structure properties of molecules on near-term noisy intermediate-scale quantum devices. In this work, we generalize the VQE algorithm for simulating periodic systems. However, the numerical study of a one-dimensional (1D) infinite hydrogen chain using existing VQE algorithms shows a remarkable deviation of the ground-state energy with respect to the exact full configuration interaction (FCI) result. Here, we present two schemes to improve the accuracy of quantum simulations for periodic systems. The first one is a modified VQE algorithm, which introduces a unitary transformation of Hartree-Fock orbitals to avoid the complex wave function. The second one is combining VQE with the quantum subspace expansion approach (VQE/QSE). Numerical benchmark calculations demonstrate that both of the two schemes provide an accurate description of the potential energy curve of the 1D hydrogen chain. In addition, excited states computed with the VQE/QSE approach also agree very well with FCI results.

DOI
10.1021/acs.jctc.0c00881

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Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.
A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational quantum eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called ``Deep VQE'', which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the technique of coarse graining [K. Fujii, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, arXiv:2007.10917]. In this paper, we extend the original proposal of Deep VQE to obtain the excited states and apply it to quantum chemistry calculation of a periodic material, which is one of the most impactful applications of the VQE. We first propose a modified scheme to construct quantum states for coarse graining in Deep VQE to obtain the excited states. We also present a method to avoid a problem of meaningless eigenvalues in the original Deep VQE without restricting variational quantum states. Finally, we classically simulate our modified Deep VQE for quantum chemistry calculation of a periodic hydrogen chain as a typical periodic material. Our method reproduces the ground-state energy and the first-excited-state energy with the errors up to O(1)O(1)%O(1) despite the decrease in the number of qubits required for the calculation by two or four compared with the naive VQE. Our result will serve as a beacon for tackling quantum chemistry problems with classically-intractable sizes by smaller quantum devices in the near future.
Accurate prediction of binding energies between drug candidates and target molecules is essential in computational drug discovery, as stronger (more negative) binding energies suggest higher efficacy. This energy corresponds to the ground state of the molecular system and can be computed using methods such as Full Configuration Interaction (FCI), Hartree-Fock (HF), Variational Quantum Eigensolver (VQE), and Quantum Phase Estimation (QPE). While FCI offers benchmark accuracy, its computational cost renders it impractical for larger systems. HF scales efficiently but at the expense of precision. QPE, though theoretically robust, is currently constrained by hardware limitations. VQE emerges as a practical alternative for Noisy Intermediate-Scale Quantum (NISQ) devices, balancing accuracy and feasibility. Nevertheless, quantum noise significantly affects VQE performance. To address this, we incorporate Zero Noise Extrapolation (ZNE) to mitigate errors and improve energy estimates. This study compares these approaches and demonstrates the efficacy of the VQEZNE framework in generating reliable binding energy calculations for drug design applications.
We demonstrate the use of the Variational Quantum Eigensolver (VQE) to simulate solid state crystalline materials. We adapt the Unitary Coupled Cluster ansatz to periodic boundary conditions in real space and momentum space representations and directly map complex cluster operators to a quantum circuit ansatz to take advantage of the reduced number of excitation operators and Hamiltonian terms due to momentum conservation. To further reduce required quantum resources, such as the number of UCCSD amplitudes, circuit depth, required number of qubits and number of measurement circuits, we investigate a translational Quantum Subspace Expansion method (TransQSE) for the localized representation of the periodic Hamiltonian. Additionally, we also demonstrate an extension of the point group symmetry based qubit tapering method to periodic systems. We compare accuracy and computational costs for a range of geometries for 1D chains of dimerized hydrogen, helium and lithium hydride with increasing number of momentum space grid points and also demonstrate VQE calculations for 2D and 3D hydrogen and helium lattices. Our presented strategies enable the use of near-term quantum hardware to perform solid state simulation with variational quantum algorithms.
Abstract A procedure for defining virtual spaces, and the periodic one-electron and two-electron integrals, for plane-wave second quantized Hamiltonians has been developed and demonstrated using full configuration interaction (FCI) simulations and variational quantum eigensolver (VQE) circuits on Quantinuum's ion trap quantum computers accessed through Microsoft's Azure Quantum service. This work is an extension to periodic systems of a new class of algorithms in which the virtual spaces were generated by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. In this extension, the integration of the first Brillouin zone is automatically incorporated into the two-electron integrals. With these procedures we have been able to derive virtual spaces, containing only a few orbitals, that were able to capture a significant amount of correlation. The focus in this manuscript is on comparing the simulations of small molecules calculated with plane-wave basis sets with large periodic unit cells at the Γ-point, including images, to results for plane-wave basis sets with aperiodic unit cells. The results for this approach were promising as we were able to obtain good agreement between periodic and aperiodic results for an LiH molecule. Simulations performed on the Quantinuum H1-1 quantum computer were able to produce surprisingly good energies, reproducing the FCI values for the 1 COVO Hamiltonian to within 11 milliHartree (6.9 kcal/mol), when corrected for noise.