Density-functional theory (DFT) has revolutionized computer simulations in chemistry and material science. A faithful implementation of the theory requires self-consistent calculations. However, this effort involves repeatedly diagonalizing the Hamiltonian, for which a classical algorithm typically requires a computational complexity that scales cubically with respect to the number of electrons. This limits DFT's applicability to large-scale problems with complex chemical environments and microstructures. This article presents a quantum algorithm that has a linear scaling with respect to the number of atoms, which is much smaller than the number of electrons. Our algorithm leverages the quantum singular value transformation (QSVT) to generate a quantum circuit to encode the density-matrix, and an estimation method for computing the output electron density. In addition, we present a randomized block coordinate fixed-point method to accelerate the self-consistent field calculations by reducing the number of components of the electron density that needs to be estimated. The proposed framework is accompanied by a rigorous error analysis that quantifies the function approximation error, the statistical fluctuation, and the iteration complexity. In particular, the analysis of our self-consistent iterations takes into account the measurement noise from the quantum circuit. These advancements offer a promising avenue for tackling large-scale DFT problems, enabling simulations of complex systems that were previously computationally infeasible.

翻译：密度泛函理论（DFT）彻底革新了化学和材料科学中的计算机模拟。该理论的高保真实现需要自洽计算。然而，这一过程涉及反复对角化哈密顿量，经典算法通常需要计算复杂度随电子数呈立方标度。这限制了DFT在具有复杂化学环境和微结构的大规模问题中的适用性。本文提出一种量子算法，其计算复杂度随原子数线性标度，而原子数远小于电子数。我们的算法利用量子奇异值变换（QSVT）生成编码密度矩阵的量子线路，并发展了一种估计输出电子密度的方法。此外，我们提出一种随机分块坐标不动点方法，通过减少需要估计的电子密度分量数量来加速自洽场计算。该框架附有严格的误差分析，量化了函数逼近误差、统计波动和迭代复杂度。特别地，我们对自洽迭代的分析考虑了来自量子线路的测量噪声。这些进展为攻克大规模DFT问题提供了有前景的途径，使得此前计算不可行的复杂系统模拟成为可能。