Let $A$ be a sparse Hermitian matrix, $f(x)$ be a univariate function, and $i, j$ be two indices. In this work, we investigate the query complexity of approximating $\bra{i} f(A) \ket{j}$. We show that for any continuous function $f(x):[-1,1]\rightarrow [-1,1]$, the quantum query complexity of computing $\bra{i} f(A) \ket{j}\pm \varepsilon/4$ is lower bounded by $\Omega(\widetilde{\deg}_\varepsilon(f))$. The upper bound is at most quadratic in $\widetilde{\deg}_\varepsilon(f)$ and is linear in $\widetilde{\deg}_\varepsilon(f)$ under certain mild assumptions on $A$. Here the approximate degree $\widetilde{\deg}_\varepsilon(f)$ is the minimum degree such that there is a polynomial of that degree approximating $f$ up to additive error $\varepsilon$ in the interval $[-1,1]$. We also show that the classical query complexity is lower bounded by $\widetilde{\Omega}(2^{\widetilde{\deg}_{2\varepsilon}(f)/6})$. Our results show that the quantum and classical separation is exponential for any continuous function of sparse Hermitian matrices, and also imply the optimality of implementing smooth functions of sparse Hermitian matrices by quantum singular value transformation. The main techniques we used are the dual polynomial method for functions over the reals, linear semi-infinite programming, and tridiagonal matrices.

翻译：令 $A$ 为稀疏埃尔米特矩阵，$f(x)$ 为单变量函数，$i, j$ 为两个指标。本文研究 $\bra{i} f(A) \ket{j}$ 近似计算的查询复杂度。我们证明：对于任意连续函数 $f(x):[-1,1]\rightarrow [-1,1]$，计算 $\bra{i} f(A) \ket{j}\pm \varepsilon/4$ 的量子查询复杂度下界为 $\Omega(\widetilde{\deg}_\varepsilon(f))$。上界至多为 $\widetilde{\deg}_\varepsilon(f)$ 的二次方，且在 $A$ 满足某些温和假设时与 $\widetilde{\deg}_\varepsilon(f)$ 呈线性关系。此处近似度 $\widetilde{\deg}_\varepsilon(f)$ 定义为存在一个该次数的多项式在区间 $[-1,1]$ 内以加性误差 $\varepsilon$ 逼近 $f$ 的最小次数。我们还证明经典查询复杂度下界为 $\widetilde{\Omega}(2^{\widetilde{\deg}_{2\varepsilon}(f)/6})$。我们的结果表明：对于稀疏埃尔米特矩阵的任意连续函数，量子与经典复杂度之间呈指数级分离，同时佐证了通过量子奇异值变换实现稀疏埃尔米特矩阵光滑函数的最优性。本文主要采用的技术包括实数域上的对偶多项式方法、线性半无限规划以及三对角矩阵。