We investigate the dividing line between classical and quantum computational power in estimating properties of matrix functions. More precisely, we study the computational complexity of two primitive problems: given a function $f$ and a Hermitian matrix $A$, compute a matrix element of $f(A)$ or compute a local measurement on $f(A)|0\rangle^{\otimes n}$, with $|0\rangle^{\otimes n}$ an $n$-qubit reference state vector, in both cases up to additive approximation error. We consider four functions -- monomials, Chebyshev polynomials, the time evolution function, and the inverse function -- and probe the complexity across a broad landscape covering different problem input regimes. Namely, we consider two types of matrix inputs (sparse and Pauli access), matrix properties (norm, sparsity), the approximation error, and function-specific parameters. We identify BQP-complete forms of both problems for each function and then toggle the problem parameters to easier regimes to see where hardness remains, or where the problem becomes classically easy. As part of our results we make concrete a hierarchy of hardness across the functions; in parameter regimes where we have classically efficient algorithms for monomials, all three other functions remain robustly BQP-hard, or hard under usual computational complexity assumptions. In identifying classically easy regimes, among others, we show that for any polynomial of degree $\mathrm{poly}(n)$ both problems can be efficiently classically simulated when $A$ has $O(\log n)$ non-zero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP-complete in the sparse access model even for constant row sparsity, whereas the stated Pauli access efficiently constructs sparse access with row sparsity $O(\log n)$. Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.

翻译：我们研究了在估计矩阵函数性质时经典与量子计算能力之间的分界线。具体而言，我们探究了两个基本问题的计算复杂性：给定函数$f$和埃尔米特矩阵$A$，计算$f(A)$的矩阵元素，或计算对$f(A)|0\rangle^{\otimes n}$的局部测量值（其中$|0\rangle^{\otimes n}$为$n$量子比特参考态向量），两种情形均要求达到加性近似误差。我们考察了四个函数——单项式、切比雪夫多项式、时间演化函数及逆函数——并在涵盖不同问题输入范围的广阔参数空间中系统探究其复杂性。具体而言，我们考虑了两类矩阵输入（稀疏访问与泡利访问）、矩阵性质（范数、稀疏度）、近似误差以及函数特定参数。我们确定了每个函数对应两种问题的BQP完全形式，随后通过调节问题参数至更简易区域，以考察计算困难性是否持续存在，或问题是否退化为经典易解情形。作为研究成果的一部分，我们明确了不同函数间的困难度层级结构：在针对单项式存在经典高效算法的参数区域中，其余三个函数仍保持强BQP困难性，或在常规计算复杂性假设下保持困难性。在界定经典易解区域方面，我们特别证明：当$A$在泡利基中具有$O(\log n)$个非零系数时，对于任意$\mathrm{poly}(n)$次多项式，两个问题均可被经典高效模拟。这一结论与稀疏访问模型中即使行稀疏度为常数时问题仍为BQP完全的事实形成鲜明对比，而上述泡利访问模型可高效构造行稀疏度为$O(\log n)$的稀疏访问。本研究为线性代数基础任务提供了量子与经典高效算法的系统汇编。