Let $A$ be an $s$-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 $\widetilde{\Omega}((s/2)^{(\widetilde{\deg}_{2\varepsilon}(f)-1)/6})$ for any $s\geq 4$. 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：设 $A$ 为 $s$-稀疏埃尔米特矩阵，$f(x)$ 为单变量函数，$i, j$ 为两个指标。本文研究逼近 $\bra{i} f(A) \ket{j}$ 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 $\widetilde{\deg}_\varepsilon(f)$ 的二次函数，且在 $A$ 的某些温和假设下为 $\widetilde{\deg}_\varepsilon(f)$ 的线性函数。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 $[-1,1]$ 内与 $f$ 的加法误差不超过 $\varepsilon$ 的最小次数。我们还证明，对于任意 $s\geq 4$，经典查询复杂度下界为 $\widetilde{\Omega}((s/2)^{(\widetilde{\deg}_{2\varepsilon}(f)-1)/6})$。研究结果表明：对于稀疏埃尔米特矩阵的任意连续函数，量子与经典复杂度存在指数级分离，同时揭示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。本文采用的主要技术包括实数域函数的对偶多项式方法、线性半无限规划以及三对角矩阵。