Ground-State Energy and Related Properties Estimation in Quantum Chemistry with Linear Dependence on the Number of Atoms

At the heart of quantum chemistry and materials science lies the critical task of estimating ground-state properties. We present a quantum algorithm for this task by quantizing the density-functional theory (DFT). A key aspect of imple-menting DFT faithfully is the requirement for self-consistent calculations, which involve repeated di-agonalizations of the Hamiltonian. This procedure, however, creates a significant bottleneck, as a classical algorithm generally demands a computational complexity that grows cubically with the number of electrons, restricting the scalability of DFT for tackling large-scale problems that involve complex chemical environments and microstructures. This article presents the first quantum algorithm that has provided substantial speedup for the ground state computation, by improving the complexity to one with a linear scaling with the number of atoms. The algorithm leverages the exponential speedup by the quantum singular value transformation to generate a quantum circuit to encode the density-matrix, followed by an efficient estimation method for the output electron density, which constitutes a simple hybrid approach for achieving self-consistency. Moreover, the algorithm produces the ground state Hamiltonian, from which the ground state energy and band structures can be efficiently computed. The proposed framework is accompanied by a rigorous error analysis that establishes the convergence and quantifies various sources of error and the overall computational complexity. The combination of effi-ciency and precision opens new avenues for exploring large-scale physical systems.