We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. The spectral sum of an PSD matrix $A$, for a function $f$, is defined as $ \text{Tr}[f(A)] = \sum_j f(\lambda_j)$, where $\lambda_j$ are the eigenvalues of $A$. Typical examples of spectral sums are the von Neumann entropy, the trace of $A^{-1}$, the log-determinant, and the Schatten $p$-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees within a graph.

翻译：我们提出了新的量子算法，用于估计半正定（PSD）矩阵的谱求和。对于函数$f$，PSD矩阵$A$的谱求和定义为$ \text{Tr}[f(A)] = \sum_j f(\lambda_j)$，其中$\lambda_j$是$A$的特征值。谱求和的典型示例包括冯·诺依曼熵、$A^{-1}$的迹、对数行列式以及Schatten $p$-范数（后者不要求矩阵为PSD）。当前最先进的经典随机算法在估计这些量时，其运行时间至少与矩阵非零元素数目成线性关系，且与估计误差的平方成正比。假设能够访问矩阵的块编码，我们的算法在矩阵规模上呈亚线性，且对其他参数（如条件数和近似误差）的依赖至多为二次关系，因此可与文献中提出的多数经典随机和分布式算法相竞争，并多项式级地改进用于解决同类问题的其他量子算法的运行时间。我们还展示了本工作中提出的算法和技术如何应用于谱图理论中的三个问题：近似图中三角形数量、有效电阻以及生成树数量。