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Quantum Computation of Electronic Transitions using a Variational Quantum Eigensolver

Robert M. Parrish, Edward G. Hohenstein, Peter L. McMahon, Todd J. Martinez

arXiv:1901.01234v2quant-ph

TL;DR

The paper tests MC-VQE on an exciton system where CIS can produce qualitatively incorrect excitation energies, oscillator strengths, and state characters. MC-VQE closely reproduces FCI, with excitation-energy errors generally at most 0.01 eV and oscillator-strength errors generally at most 0.1.

Abstract

from arXiv · show

We develop an extension of the variational quantum eigensolver (VQE) algorithm - multistate, contracted VQE (MC-VQE) - that allows for the efficient computation of the transition energies between the ground state and several low-lying excited states of a molecule, as well as the oscillator strengths associated with these transitions. We numerically simulate MC-VQE by computing the absorption spectrum of an ab initio exciton model of an 18-chromophore light-harvesting complex from purple photosynthetic bacteria.

HAMILTONIAN MANIPULATION

The ab initio exciton Hamiltonian represents monomer energies and electrostatic interactions in Pauli-operator form. Its dominant terms are the monomer energy differences, while transition and intermonomer couplings encode excitation and interaction effects.

HAMILTONIAN MANIPULATION

  • The model uses two electronic states per monomer and converts one- and two-body Hamiltonian terms into sums of tensor products of Pauli operators.The one-body terms include local ground/excited energies and intramonomer couplings; two-body terms encode interactions between monomers.Source: HAMILTONIAN MANIPULATION · p. 7
  • The one-body matrix elements describe monomer adiabatic energy levels and transition terms, while two-body elements describe electrostatic interactions between densities and transition densities.These quantities can be computed with polynomial classical electronic-structure methods for monomers with several hundred atoms.Source: HAMILTONIAN MANIPULATION · p. 7
  • The two-body contribution requires separate algebraic reductions of its terms before the full Hamiltonian can be written in Pauli notation.The passage distinguishes the reductions for terms 1, 5, 6, and 9 from those for terms 2, 3, 7, and 8.Source: HAMILTONIAN MANIPULATION · pp. 7–8
  • The Hamiltonian includes hole, particle, and transition operators, with H = (I + Z)/2 and P = (I − Z)/2 defining hole and particle counting.The identity and Z operators satisfy I = H + P and Z = H − P.Source: HAMILTONIAN MANIPULATION · p. 7
  • ZA terms are typically largest at a few eV, while XA and XXAB terms are commonly next largest at 10^-3–10^-2 au.These are rough estimates that may differ across systems.Source: HAMILTONIAN MANIPULATION · p. 8

MC-VQE TECHNICAL DETAILS

MC-VQE prepares CIS contracted reference states, applies an orthogonal entangler, and classically optimizes its parametrization. The technical results show that logical parametrization and optimizer choice materially affect convergence, even when parametrizations cover the same SO(4) space.

MC-VQE TECHNICAL DETAILS

  • MC-VQE prepares N + 1 orthonormal CIS states: one ground configuration and N singly excited configurations that may mix.The CIS coefficients are obtained classically, and the reference states are prepared by a recursive circuit construction.Source: MC-VQE TECHNICAL DETAILS · p. 9
  • CIS state preparation requires recursive angle evaluation, and the final angle is resolved only up to a ±1 phase that must be checked explicitly.The phase check produces a relative error of 0 or 2 in the stated test.Source: MC-VQE TECHNICAL DETAILS · pp. 9–10
  • The global entangler is constructed from two-body entanglers restricted to SO(4), with additional layers or longer-range nearest-neighbor entanglers available for greater variational flexibility.The total real entangler can be restricted to SO(2^N) without loss for the real orthonormal eigenfunctions considered.Source: MC-VQE TECHNICAL DETAILS · p. 10
  • The two-body entangler uses SO(4), the group of real orthogonal 4 × 4 matrices with determinant +1, providing all two-body entanglers needed for the task.SO(4) has six real parameters, and the entangler can be implemented through equivalent logical parametrizations.Source: MC-VQE TECHNICAL DETAILS · pp. 10–11
  • The second SO(4) parametrization is sometimes easier to optimize with finite-difference gradient descent from zero entanglement, while tightly converged L-BFGS gives similar excitation energies and oscillator strengths for both parametrizations.The primary calculations use the second logical parametrization with L-BFGS and finite-difference gradients from all-zero angles.Source: MC-VQE TECHNICAL DETAILS · p. 11
  • Direct steepest-descent optimization of another SO(4) parametrization reached a local minimum that was not qualitatively superior to CIS, whereas L-BFGS converged to the desired solution in about twice as many iterations.The authors identify logical parametrization and robust convergence as subjects for future work.Source: MC-VQE TECHNICAL DETAILS · p. 11

COMPUTATIONAL DETAILS

The computational setup builds the B850 exciton model from isolated-monomer quantum chemistry and dipole-dipole couplings, then evaluates energies and oscillator strengths with MC-VQE. The implementation records the geometry, optimized parameters, and convergence data needed for replication.

COMPUTATIONAL DETAILS

  • The B850 calculation uses an N = 18 BChl-a LH2 ring, with monomer energies and dipoles computed using ωPBE(ω = 0.3)/6-31G* and excited states from TDA-TD-DFT.The S1 and S0–S1 transition dipoles are computed without orbital response.Source: COMPUTATIONAL DETAILS · pp. 11–12
  • The computational record includes the code revision, optimized MC-VQE parameters, L-BFGS history, and converged-solution data for the full B850 system.These materials are supplied in the supplemental data packet and associated output files.Source: COMPUTATIONAL DETAILS · pp. 11–12
  • Two-body interaction matrix elements are computed from dipole-dipole interactions using monomer dipoles, displacement-direction normals, and center-of-mass distances.The relevant geometric quantities are the dipole vector μA, normal vector nAB, and separation rAB.Source: COMPUTATIONAL DETAILS · p. 12
  • Dipole oscillator strengths are computed from energy gaps and transition dipole moments between approximate eigenstates.The dipole operators are represented as sums of one-body Pauli operators.Source: COMPUTATIONAL DETAILS · p. 12

ADDITIONAL CASE STUDY: BCHL-A H-AGGREGATE STACK

In the N = 8 BChl-a H-aggregate stack, MC-VQE closely reproduces FCI absorption spectra despite poor CIS reference states. The case study shows reliable excitation energies, oscillator strengths, and state characters for a system with significant multi-excitonic character.

Computational setup

  • The test uses an ab initio exciton model for eight relaxed BChl-a units, with monomer treatments from TDA-TD-DFT and dipole-dipole couplings.The lowest nine states and eight ground-to-excited oscillator strengths were computed using FCI, CIS, and MC-VQE.Source: ADDITIONAL CASE STUDY: BCHL-A… · p. 12

Physical setting

  • The aligned stack exhibits H-aggregate behavior, with blue-shifted monomer states, attenuated low-state oscillator strengths, and strong influence from double and higher excitations.These features make CIS a poor starting approximation for the adiabatic states.Source: ADDITIONAL CASE STUDY: BCHL-A… · p. 12

Results

  • CIS excitation energies are blue-shifted by 0.1–0.5 eV, while oscillator-strength relative errors approach 100% for states with significant strengths.The CIS absorption profile is therefore qualitatively and quantitatively incorrect for this stack.Source: ADDITIONAL CASE STUDY: BCHL-A… · p. 13
  • MC-VQE recovers state characters closely, with FCI–MC-VQE fidelity of 0.995 compared with 0.688 for FCI–CIS on the brightest transition.The apparent CIS–FCI oscillator-strength agreement for S0-to-S7 is accidental because the corresponding excited-state populations differ markedly.Source: ADDITIONAL CASE STUDY: BCHL-A… · p. 13
  • MC-VQE excitation energies differ from FCI by at most 0.01 eV and oscillator strengths by at most 0.1, producing a generally visually indistinguishable absorption spectrum.The comparison uses the lowest eight transitions and Lorentzian broadening with δ = 0.15 eV.Source: ADDITIONAL CASE STUDY: BCHL-A… · p. 13, Figure 3 · p. 13

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