Quantum and classical query complexities of functions of matrices

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. As another hardness result, we show that entry estimation problem (i.e., deciding $\bra{i} f(A) \ket{j}\geq \varepsilon$ or $\bra{i} f(A) \ket{j}\leq -\varepsilon$) is BQP-complete for any continuous function $f(x)$ as long as its approximate degree is large enough. The main techniques we used are the dual polynomial method for functions over the reals, linear semi-infinite programming, and tridiagonal matrices.