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Physical Review A

Calculation of vibrational eigenenergies on a quantum computer: Application to the Fermi resonance in CO 2

Erik Lötstedt, Kaoru Yamanouchi, 敬史 土屋, Yutaka Tachikawa

Published Jun 16, 2021• 19 citations
Publisher

Abstract

We apply a modified version of the multistate contracted variational quantum eigensolver method to calculate vibrational eigenstates of ${\mathrm{CO}}_{2}$ on a quantum computer. A two-mode model of ${\mathrm{CO}}_{2}$ is employed, and the vibrational wave function is expanded using three harmonic-oscillator basis functions for each mode. The wave functions are mapped to four qubits by a compact mapping method. The Hamiltonian matrix elements are evaluated on a simulator including noise and on a quantum computer available at IBM Quantum, while the Hamiltonian matrix is diagonalized on a classical computer. We propose an error mitigation method by which the shift of the numerical values of the matrix elements originating from the noise can be corrected, and examine the dependence of the statistical uncertainties on the number of executions of each quantum circuit. We find that, at about $8\ifmmode\times\else\texttimes\fi{}{10}^{6}$ executions, the energy eigenvalues of the Fermi resonance states in ${\mathrm{CO}}_{2}$ can be obtained with an uncertainty within 1 ${\mathrm{cm}}^{\ensuremath{-}1}$.

DOI
10.1103/physreva.103.062609

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Abstract Quantum computing has recently been emerging in theoretical chemistry as a realistic avenue meant to offer computational speedup to challenging eigenproblems in the context of strongly-correlated molecular systems or extended materials. Most studies so far have been devoted to the quantum treatment of electronic structure, for which the transformation of fermionic operators into the qubit space is quite transparent. In contrast, only a few were directed to the quantum treatment of vibrational structure, which at the moment remains not devoid of unknowns. When simulating an anharmonic vibrational mode under bosonic second quantization, the biggest problem, which we analyze in detail in the present work, is the disruption of the resolution of the identity when truncating the infinite harmonic-oscillator basis set of primitive modals and how this alters the canonical commutation relation. This may lead to serious incorrectness in the evaluation of the Hamiltonian matrix elements when assembling them from finite matrix products, which eventually entail a nonvariational behavior with respect to the finite size of the computational basis. As we show here, a simple cure occurs to be obtained upon using Wick's normal order, for reasons that are not standardly evoked. We also provide a detailed comparison between the boson-to-qubit unary and binary mappings under two different representations: one for the bosonic ladder operators within the harmonic-oscillator primitive basis, and one for the so-called n-mode representation within any type of computational basis. In addition, we discuss the impact of choosing an adequate primitive basis set in terms of quantum computing with respect to its variational convergence efficiency (number of basis functions, hence of qubits) and as regards the magnitude of the 1-norm of the encoded Hamiltonian (a measure of the computational complexity of the quantum algorithm). Such fundamental aspects are illustrated numerically on a one-dimensional anharmonic Hamiltonian model corresponding to a symmetric double-well potential, of interest both for vibrational spectroscopy and chemical reactivity, and which is a challenging situation for numerical convergence due to fine tunneling splitting.
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In this work, we present a comprehensive comparative study between two different Ansatz quantum circuits , specifically the Unitary Vibrational Coupled Cluster (UVCC) and the Compact Heuristic Circuit (CHC) with use in computational chemistry . By incorporating these circuits into the Variational Quantum Eigensolver (VQE), we aim to find the vibrational ground state energy of small molecules. Our results, obtained using both the UVCC and CHC ansatzes with and without quantum hardware noise, are benchmarked against classical computational methods for energy state determination. We emphasize the efficiency and accuracy of the CHC Ansatz , which significantly reduces circuit complexity without sacrificing the fidelity of the results. This characteristic demonstrates the potential of the CHC Ansatz for scalable quantum chemistry applications, particularly in the Noisy Intermediate-Scale Quantum (NISQ) era. Furthermore, we employ the CHC Ansatz in conjunction with the Variational Quantum Deflation (VQD) algorithm to determine the excited vibrational state energies, which is compared against the reliable method of quantum Equation of Motion(qEOM). Through this comparative study, we aim to provide valuable insights into the practical applications of these Ansatz in quantum simulations of molecular systems, highlighting their respective advantages and potential for future quantum chemistry research.
Abstract Quantum computers have the potential to help solve a range of physics and chemistry problems, but noise in quantum hardware currently limits our ability to obtain accurate results from the execution of quantum-simulation algorithms. Various methods have been proposed to mitigate the impact of noise on variational algorithms, including several that model the noise as damping expectation values of observables. In this work, we benchmark various methods, including a new method proposed here. We compare their performance in estimating the ground-state energies of several instances of the 1D mixed-field Ising model using the variational-quantum-eigensolver algorithm with up to 20 qubits on two of IBM’s quantum computers. We find that several error-mitigation techniques allow us to recover energies to within 10% of the true values for circuits containing up to about 25 ansatz layers, where each layer consists of CNOT gates between all neighboring qubits and Y -rotations on all qubits.