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Density-functional theory (DFT) has revolutionized computer simulations inchemistry and material science. A faithful implementation of the theoryrequires self-consistent calculations. However, this effort involves repeatedlydiagonalizing the Hamiltonian, for which a classical algorithm typicallyrequires a computational complexity that scales cubically with respect to thenumber of electrons. This limits DFT's applicability to large-scale problemswith complex chemical environments and microstructures. This article presents aquantum algorithm that has a linear scaling with respect to the number ofatoms, which is much smaller than the number of electrons. Our algorithmleverages the quantum singular value transformation (QSVT) to generate aquantum circuit to encode the density-matrix, and an estimation method forcomputing the output electron density. In addition, we present a randomizedblock coordinate fixed-point method to accelerate the self-consistent fieldcalculations by reducing the number of components of the electron density thatneeds to be estimated.  The proposed framework is accompanied by a rigorous error analysis thatquantifies the function approximation error, the statistical fluctuation, andthe iteration complexity. In particular, the analysis of our self-consistentiterations takes into account the measurement noise from the quantum circuit.These advancements offer a promising avenue for tackling large-scale DFTproblems, enabling simulations of complex systems that were previouslycomputationally infeasible.

Implementation of the Density-functional Theory on Quantum Computers with Linear Scaling with respect to the Number of Atoms