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System-bath model for quantum chemistry

Dmitry S. Golubev, Reza G. Shirazi, Vladimir V. Rybkin, Benedikt M. Schoenauer, Peter Schmitteckert, Michael Marthaler

Abstract

We propose an approximate mapping of a molecular Hamiltonian to a Hamiltonian of qubits, which allows for high accuracy quantum chemistry calculations of vertical excitation energies of some molecules. The mapping is based on separating of a very small active space of only two orbitals and on modeling the electronic excitations in the remaining orbitals by a set of qubits or, equivalently, by a set of oscillators. This approach is inspired by the Random Phase Approximation (RPA), in which the excitations of electron gas are described by bosonic degrees of freedom. As a result, the Hamiltonian of the molecule is reduced to that of a system-bath model. The "system" part of the Hamiltonian describes the two molecular orbitals -- the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) -- which are populated by two electrons. Two qubits are sufficient to encode the Hamiltonian of such a system. The "bath" consists of oscillators or, equivalently, of two level systems with each of them corresponding to an electron excitation from a doubly occupied orbital below the Fermi level to an empty orbital above the Fermi level. We hope that this mapping can inspire new approaches and algorithms aimed at calculating excitation energies of molecules on near term quantum computers.

System-bath model for quantum chemistry

Abstract

We propose an approximate mapping of a molecular Hamiltonian to a Hamiltonian of qubits, which allows for high accuracy quantum chemistry calculations of vertical excitation energies of some molecules. The mapping is based on separating of a very small active space of only two orbitals and on modeling the electronic excitations in the remaining orbitals by a set of qubits or, equivalently, by a set of oscillators. This approach is inspired by the Random Phase Approximation (RPA), in which the excitations of electron gas are described by bosonic degrees of freedom. As a result, the Hamiltonian of the molecule is reduced to that of a system-bath model. The "system" part of the Hamiltonian describes the two molecular orbitals -- the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) -- which are populated by two electrons. Two qubits are sufficient to encode the Hamiltonian of such a system. The "bath" consists of oscillators or, equivalently, of two level systems with each of them corresponding to an electron excitation from a doubly occupied orbital below the Fermi level to an empty orbital above the Fermi level. We hope that this mapping can inspire new approaches and algorithms aimed at calculating excitation energies of molecules on near term quantum computers.
Paper Structure (21 sections, 141 equations, 4 figures, 5 tables)

This paper contains 21 sections, 141 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Hamiltonian in simple terms
  3. Model
  4. Hamiltonian of a molecule
  5. System-bath model
  6. Hamiltonian of a molecule in terms of qubit operators
  7. Static screening approximation
  8. Mixed approximation
  9. Results and discussion
  10. Acknowledgment
  11. Qubit representation of System Hamiltonian
  12. Hamiltonian of the environment, transitions between the environment orbitals
  13. Hamiltonian of a molecule with the singlet ground state of $H_{\rm sys}$
  14. Approximating interaction between the HOMO/LUMO and the environment orbitals
  15. Summary: Hamiltonian of a molecule with the singlet ground state of $H_{\rm sys}$
  16. ...and 6 more sections

Figures (4)

  • Figure 1: Possible transitions between the environment orbitals (arrows in the left side of both figures) and between the environment orbitals and the HOMO/LUMO orbitals (arrows in the right side of both figures). Each of the transitions corresponds to an oscillator in the bath. Left figure refers to the molecules with the singlet ground state of the isolated HOMO/LUMO orbitals, and the right figure --- to the molecules with the triplet ground state of the isolated HOMO/LUMO orbitals.
  • Figure 2: Discrete transverse coupling spectrum associated with the system--bath mapping for thiophene molecule. We plot the quantity $S(\omega)=\sum_{m\alpha}(g_{12}^{m\alpha})^2\,\delta\!\left(\omega-\Omega_{m\alpha}\right)$, i.e., a spectral-density-like representation of the mode frequencies $\Omega_{m\alpha}$ and the transverse couplings $g_{12}^{m\alpha}$. $\delta$-functions are replaced by Lorentzian peaks with the half-width 0.02 eV. $S(\omega)$ is directly analogous to the spectral density in the spin--boson models, which motivates importing spin--boson discretization/encoding techniques Miessen2021Burger2022Tudorovskaya2024Walters2024LasHeras2015 into the present setting.
  • Figure 3: Vertical triplet-singlet gaps $E_{\rm T}-E_{\rm S}$ (eV) for three selected molecules: cyclopentadiene with the total number of environment qubits $N_{\rm RPA}=3977$; pyrrole with the total number of environment qubits $N_{\rm RPA}=3815$; and thiophene with the total number of environment qubits $N_{\rm RPA}=4575$. Blue dots indicate the results of the Quantum Solver simulations with the number of qubits in the environment in the range $0\leq N_q \leq 62$ and with no more than 4 spin excitations, orange dots show the $E_{\rm T}-E_{\rm S}$ gaps computed for $63\leq N_q \leq 126$ and with no more than 2 spin excitations, orange dashed lines show the values obtained within the static approximation of Sec. \ref{['Sec_static']}, and blue dotted lines are the MR-AQCC values. Blue and orange dots are computed within the mixed model of Sec. \ref{['mixed']}, in which $N_q$ strongest coupled bath qubits are kept in the environment and the remaining $N_{\rm RPA}-N_q$ weaker coupled bath qubits are absorbed in the Lamb shift, which results in the renormalization of the system parameters (\ref{['params_2']}).
  • Figure 4: Modeling of the interaction terms (\ref{['Vint']}). (a) The terms $V_{\rm ex}+V_{\rm pair}$, defined in Eqs. (\ref{['V_ex']},\ref{['V_pair']}), are replaced by additional oscillators in the environment, indicated by arrows, which correspond to the transitions between HOMO/LUMO and environment orbitals. (b) The term $V_1$ (\ref{['V1']}) leads to interaction between the oscillators describing the transitions between the environment orbitals (left arrow) and the oscillators originating from transitions between HOMO/LUMO and environment orbitals. Dashed lines with arrows indicate this interaction. (c) The Coulomb interaction term $V_{\rm C}$ (\ref{['V_C']}) leads to the interaction between the HOMO/LUMO orbitals and the transitions involving the environment orbitals. (d) The term $V_3$ (\ref{['V3']}) describes the interaction between the HOMO/LUMO orbitals and the oscillators coming from the terms $V_{\rm ex} + V_{\rm pair}$.