A quantum computing implementation of nuclearelectronic orbital (NEO) theory: Toward an exact pre-Born–Oppenheimer formulation of molecular quantum systems

Abstract

Nuclear quantum phenomena beyond the Born–Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron–nuclear quantum dynamics, it is recognized that these problems scale exponentially with system size on classical processors and, therefore, may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron–nuclear problem on near-term quantum computers, based upon the Nuclear–Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework, thereby reducing the Hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialization techniques, which improve convergence behavior relative to conventional initialization. These techniques are applied to H2 and malonaldehyde for which results agree with NEO full configuration interaction and NEO complete active space configuration interaction benchmarks for ground state energy to within 10−6 hartree and entanglement entropy to within 10−4. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy.

Nonadiabatic Nuclear–Electron Dynamics: A Quantum Computing Approach

“…This is obtained according to s = prefix− Tr ( ρ̂ e ln ⁡ ρ̂ e ) = prefix− Tr ( ρ̂ n ln ⁡ ρ̂ n ) where ρ̂ e = Tr n [ρ̂ e,n ] and ρ̂ n = Tr e [ρ̂ e,n ] are the reduced density matrices corresponding to electrons and nuclei respectively, and ρ̂ e,n is the full density matrix. In contrast to our previous study, the proton–electron entanglement in the Ĥ L and Ĥ R ground states is no longer zero but amounts to 0.0020 and 0.0019 for NEOUCCSDT and NEOCASCI, respectively. This is mainly due to the extension of nuclear active space for Ĥ L and Ĥ R (compared to ref ), which includes orbitals at the top of, as well as on both sides of the barrier separating the two minima.…”

“…In contrast to our previous study, the proton–electron entanglement in the Ĥ L and Ĥ R ground states is no longer zero but amounts to 0.0020 and 0.0019 for NEOUCCSDT and NEOCASCI, respectively. This is mainly due to the extension of nuclear active space for Ĥ L and Ĥ R (compared to ref ), which includes orbitals at the top of, as well as on both sides of the barrier separating the two minima. Our calculations still show increasing electron–nuclear entanglement as the proton approaches the top of the barrier.…”

“…electronic orbitals unitary coupled cluster singles, doubles, and triples (NEOUCCSDT) ansatz. 17 As a model for the demonstration of our quantum algorithm for electron−proton quantum dynamics, we consider the proton transfer process in malonaldehyde using the setup discussed in our previous publication. 17 As shown in Figure 1, the proton can be localized at two possible asymmetric equilibrium positions separated from each other by 0.42 Å (marked with blue and green spheres) as well as at the peak of the barrier in the middle (orange sphere).…”

“…It is important to stress that neither the Slater determinants nor the molecular orbitals are allowed to change during the dynamics, but only the CI coefficients C μν ( t ) evolve in time according to the time-dependent Schrödinger equation for the molecular Hamiltonian in eq . Thus, the evolution of the wave function can be expressed by the equation of motion for CI coefficients bold-italicHC = i ∂ ∂ t bold-italicC where matrix elements of H are defined through H κ λ , μ ν = ⟨ Φ κ e | ⟨ Φ λ n | Ĥ | Φ μ e ⟩ | Φ ν n ⟩ On a quantum computer, the initial NEO wave function, |Ψ( t 0 )⟩, will be efficiently approximated using the nuclear-electronic orbitals unitary coupled cluster singles, doubles, and triples (NEOUCCSDT) ansatz …”

“…These setups define three different Hamiltonians: Ĥ R 0 ( R for “Right”), Ĥ L 0 ( L for “Left”), and Ĥ M 0 ( M for “Middle”), respectively. The nuclear and electronic orbitals were optimized with NEOHF calculations using the transition state molecular scaffold of malonaldehyde and imposing C s symmetry. For protons, a split-valence double-ζ nuclear basis set composed of 2 uncontracted Cartesian S functions (DZSNB) was used.…”