The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given interval. Our quantum algorithm is based on a method that approximates the eigenvalue counts by applying the numerical contour integral and the stochastic trace estimator applied to a matrix involving resolvent matrices. As components of our algorithm, the HHL solver is applied to an augmented linear system of the resolvent matrices, and the quantum Fourier transform (QFT) is adopted to represent the operation of the numerical contour integral. To reduce the size of the augmented system, we exploit a certain symmetry of the numerical integration. We also introduce a permutation formed by CNOT gates to make the augmented system solution consistent with the QFT input. The eigenvalue count in a given interval is derived as the probability of observing a bit pattern in a fraction of the qubits of the output state.
翻译:在电子结构计算等各类科学计算中,矩阵的外观值密度在电子结构计算等不同类型科学计算中起着重要作用。在本文中,我们提出在一定间隔内计算外观值密度的量子算法。我们的量子算法基于一种方法,该方法通过应用数值等离子元集成和与溶解矩阵的矩阵应用的随机微量估计值值值值值值来接近外观值值。作为我们算法的组成部分,HHL解算法应用到一个增强的分辨率矩阵的线性系统中,而量子 Fourier变换(QFT)则被采用来代表数字等离子集成的操作。为了缩小扩展系统的大小,我们利用了数字整合的某种对称。我们还采用了由CNOT门形成的对等法,以使增强的系统解算法与QFT输入一致。在一定间隔内生成的外观值值值值值值值作为在输出状态的一小部分中观测某种微模式的概率。