Evaluating the efficiency of ground-state-preparation algorithms

In recent years, substantial research effort has been devoted to quantum algorithms for ground-state-energy estimation (GSEE) in chemistry and materials. Given the many heuristic and nonheuristic methods being developed, it is challenging to assess what combination of these methods will ultimately be used in practice. One important metric for assessing utility is the runtime, which depends on the ground-state preparation (GSP) for most GSEE algorithms. Towards assessing the utility of various combinations of GSEE and GSP methods, we asked under which conditions a GSP method should be accepted over a reference method, such as the Hartree-Fock method. We introduce a criterion for accepting or rejecting a GSP method for the purposes of GSEE. We consider different GSP methods ranging from heuristics to algorithms with provable performance guarantees and perform numerical simulations to benchmark their performance on different chemical systems, starting from small molecules like the hydrogen molecule to larger systems like jellium. In the future, this approach may be used to abandon certain variational quantum eigensolver (VQE) ansatzes and other heuristics. Yet our findings do not provide evidence against using VQE and more expensive heuristic methods, like the low-depth booster. This work sets a foundation from which to further explore the requirements to achieve quantum advantage in quantum chemistry.

Figure 1

The acceptability criteria are used to benchmark the given GSP method over the HF reference with the goal of reducing the total runtime of the GSEE algorithm.

Figure 2

Fidelity as a function of bond distance of the ${\text{H}}_2$ molecule for the HF method and the SPA algorithm as GSP methods, respectively. The value of $\alpha$ and $\beta$ is set to be $\alpha +\beta =1$, which corresponds to a GSEE method that has a more strict acceptability criterion.

Figure 3

Maximum acceptable depth $D_{max}$ of the GSP algorithm with $\gamma =1$ and $D_{\mathrm{GSEE}}=1.8\times {10}^7$ for solid-state materials as a function of the HF overlap ${\gamma }_0$.