We introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We showcase our approach by calculating the matrix elements of the exponential of a transverse-field Ising model and evaluating quantum transition amplitudes for large many-body Hamiltonians of sizes up to $2^{64} \times 2^{64}$ on a single workstation. We also discuss the application of the method to matrix inverses. We relate and compare our method to the state-of-the-art and demonstrate its advantages. We also discuss practical applications of our method.

翻译：我们采用一种方法来计算矩阵函数的个别要素。我们的技术利用新颖的系列扩展来根据即使是非常大的矩阵也能提高记忆力的矢量来采取行动矩阵函数。我们展示了我们的做法,我们计算了横贯地轴线模型指数的矩阵要素,并评价了一个工作站上大小高达2 ⁇ 64美元/乘以2 ⁇ 64美元/乘以2 ⁇ 64美元/乘以2 ⁇ 64美元的大型汉密尔顿人的量子过渡振幅。我们还讨论了对矩阵反向应用该方法的问题。我们将我们的方法与最新技术联系起来并进行比较,并展示其优点。我们还讨论了我们方法的实际应用。