Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry

Abstract

Due to intense interest in the potential applications of quantum computing, it is critical to understand the basis for potential exponential quantum advantage in quantum chemistry. Here we gather the evidence for this case in the most common task in quantum chemistry, namely, ground-state energy estimation, for generic chemical problems where heuristic quantum state preparation might be assumed to be efficient. The availability of exponential quantum advantage then centers on whether features of the physical problem that enable efficient heuristic quantum state preparation also enable efficient solution by classical heuristics. Through numerical studies of quantum state preparation and empirical complexity analysis (including the error scaling) of classical heuristics, in both ab initio and model Hamiltonian settings, we conclude that evidence for such an exponential advantage across chemical space has yet to be found. While quantum computers may still prove useful for ground-state quantum chemistry through polynomial speedups, it may be prudent to assume exponential speedups are not generically available for this problem.

Fig. 1. Ansatz state preparation and ansatz weights for model Fe-S clusters.

A Structural models of [2Fe-2S], [4Fe-4S], P-cluster, and FeMo-co. B Weight of two different types of ansatz state: largest weight determinant (Φ_D) (purple) and largest weight configuration state function (Φ_CSF) (orange) as a function of the number of metal centers in each cluster (using split-localized orbitals). P^N, P^syn, P^ox here refer to different oxidation states of the metal ions in the P-cluster. Both types of ansatz state show an exponential decrease in weight with the number of metal centers. For the [2Fe-2S] clusters, we also show results for the largest weight determinant using natural orbitals (empty symbols).

Fig. 2. Adiabatic state preparation for a model [2Fe-2S] cluster.

A Structure and simplified active space model of [2Fe-2S] cluster. B ASP time and the adiabatic estimate. We see that the ratio $T_{\mathrm{ASP}}/T_{\mathrm{ASP}}^{\mathrm{est}}$ is O(1). C Adiabatic estimates ($T_{\mathrm{ASP}}^{\mathrm{est}}$) for two families of initial Hamiltonians against the weight of the initial ground state (ϒ₀) in the final ground state (Ψ₀) (∣〈ϒ₀∣Ψ₀〉∣²), showing an inverse dependence on the initial weight. The mean-field Hamiltonians are constructed to have different Slater determinants as their ground-state, while the interacting Hamiltonians contain the full electron interaction amongst n_act orbitals. Additional discussion in Supplementary Notes 3.3 and 5.

Fig. 3. Computational complexity of classical heuristics for molecular systems.

A Energy error of a nitrogen molecule (equilibrium geometry) as a function of the level of CC approximation, against a computational time metric. Data taken from ref. , time metric defined in Supplementary Note 3.7. The observed precision cost is like poly(1/ϵ). B Cost of a state-of-the-art reduced-scaling coupled-cluster (CCSD(T)) implementation scales nearly-linearly with the system size in gapped systems, as demonstrated here for n-alkanes (C_mH_2m+2) with m = [20…120]. Size-extensivity of the coupled-cluster ansatz ensures constant error per system subunit, as illustrated in the subfigure for the error of explicitly-correlated reduced-scaling CCSD(T) (see Supplementary Note 6.1 for details) with respect to the available experimental gas-phase enthalpy of formation in the standard state for n-alkanes with m = [2…20]. C Reduced-scaling CCSD(T) implementations can be routinely applied to systems with thousands of electrons on a few computer cores, as demonstrated here for a small fragment of photosystem II.

Fig. 4. Computational complexity of classical heuristics for models of strongly correlated material systems.

A Relative energy error of a tensor network (PEPS) with respect to system sizes 3³ to 10³ for the 3D Heisenberg cube model with a bond dimension (D) of 4. In bottom panel: total computational time in seconds, divided by number of sites, as a function of system size, demonstrating poly(L) (close to linear) computational effort. B Energy convergence of PEPS with respect to the bond dimension for 2D Hubbard models at half filling (4 × 4 lattice, in top panel) and the challenging 1/8 doping point (4 × 4, 8 × 4 and 16 × 4 lattice, in bottom panel). The plots are consistent with $1/\overline{ϵ}~\text{poly}\left(D\right)$ with a weak dependence on L.