Quantum computing enhanced computational catalysis

The quantum computation of electronic energies can break the curse of dimensionality that plagues many-particle quantum mechanics. It is for this reason that a universal quantum computer has the potential to fundamentally change computational chemistry and materials science, areas in which strong electron correlations present severe hurdles for traditional electronic structure methods. Here we present a state-of-the-art analysis of accurate energy measurements on a quantum computer for computational catalysis, using improved quantum algorithms with more than an order of magnitude improvement over the best previous algorithms. As a prototypical example of local catalytic chemical reactivity we consider the case of a ruthenium catalyst that can bind, activate, and transform carbon dioxide to the high-value chemical methanol. We aim at accurate resource estimates for the quantum computing steps required for assessing the electronic energy of key intermediates and transition states of its catalytic cycle. In particular, we present quantum algorithms for double-factorized representations of the four-index integrals that can significantly reduce the computational cost over previous algorithms, and we discuss the challenges of increasing active space sizes to accurately deal with dynamical correlations. We address the requirements for future quantum hardware in order to make a universal quantum computer a successful and reliable tool for quantum computing enhanced computational materials science and chemistry, and identify open questions for further research.

Figure 1

Selected steps of the catalytic cycle elucidated in Ref. [27] on the basis of DFT calculations: intermediates and transition state structures considered for the present work are shown (Roman numerals are according to the original publication).

Figure 2

Relative DFT electronic reaction energies for the catalytic cycle obtained with a def2-TZVP basis set and three different approximate density functionals: M06-L [27], PBE, and PBE0 on M06-L/def2-SVP-optimized structures taken from Ref. [27]. For comparison, we provide relative electronic reaction energies from PBE/def2-TZVP//PBE/def2-TZVP calculations (in Pople's double slash notation, where the density functional behind the double slash is the one for which the structure was obtained) labeled as “PBE opt.”

Figure 3

Protocol of computational catalysis with the key step of quantum computing embedded in black, which is usually accomplished with traditional methods such as CASSCF, DMRG, or FCIQMC (see text for further explanation).

Figure 4

Empirical absolute error $\delta$ (in Hartree) of the ground-state electronic energy resulting from applying the two truncation schemes to the two-electron integrals of the Hamiltonian at various thresholds ${\epsilon }_{\text{co}},{\epsilon }_{\text{in}}$, evaluated from DMRG-CI calculations. The lines $\delta =\epsilon$ and $\delta =1\text{mHartree}$ are meant to guide the eye. (Left) Absolute error of the ground-state electronic DMRG-CI energy for the two truncation schemes for complex II–III with 6 active orbitals and complex IX with 16 active orbitals. (Middle) Absolute error in the ground-state electronic DMRG-CI energy of linear hydrogen chains of length 2, 4, 6, and 8 at different truncation thresholds ${\epsilon }_{\text{co}}$ for the coherent truncation scheme and (right) ${\epsilon }_{\text{in}}$ for the incoherent truncation.

Figure 5

(Left) Toffoli cost versus truncation threshold for phase estimation to a precision of 1 mHartree in the double-factorized representation of the catalytic cycle in Fig. 1 with a number of orbitals listed in Table 1. Computation time is assumed to be $10\mu \mathrm{s}$ between each Toffoli, or 100 kHz. (Right) Number of eigenvector in the double-factorized representation. The data points by Peng et al. are for three-dimensional hydrocarbons [55] with 54 orbitals.

Figure 6

(Left) Scaling of minimized Toffoli cost across all configurations of the catalytic cycle with respect to active space size, for performing phase estimation to a precision of 1 mHartree in the double-factorized representation. The fitted power-law curve has an exponent arising from the product ${\alpha }_{\text{DF}}{c}_W\sim {\alpha }_{\text{DF}}\sqrt{MN}/{\mathrm{\Delta }}_E$ in Eqs. (1) and (19) (middle) where $M$ is the number of eigenvectors and (right) the normalizing constant is ${\alpha }_{\text{DF}}$.