We study the computational complexity of estimating the normalized trace $2^{-n}Tr[f(A)]$ for a log-local Hamiltonian $A$ acting on $n$ qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions $f(x)$. We show that if $f(x)$ is a continuous function with approximate degree $Ω({\rm poly}(n))$, then estimating $2^{-n}Tr[f(A)]$ up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of $f(x)$. This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when $A$ is sparse, the classical query complexity of this problem is exponential in the approximate degree, assuming a conjectured lower bound for a trace variant of the $k$-Forrelation problem in the DQC1 query model. Together, these results identify the approximate degree as the key parameter governing the complexity of normalized trace estimation: it characterizes both the quantum complexity (via efficient DQC1 algorithms) and, conditionally, the classical hardness, yielding an exponential quantum-classical separation. Our proof develops a unified framework that cleanly combines circuit-to-Hamiltonian constructions, periodic Jacobi operators, and tools from polynomial approximation theory, including the Chebyshev equioscillation theorem.

翻译：我们研究了作用于n个量子比特的局部哈密顿量A的归一化迹$2^{-n}Tr[f(A)]$估计的计算复杂度。该问题自然出现在DQC1模型中，但此前仅对有限类函数$f(x)$理解其复杂度。我们证明：若$f(x)$是具有近似度$Ω({\rm poly}(n))$的连续函数，则在关于$f(x)$多项式逼近误差的技术条件下，以常数加法误差估计$2^{-n}Tr[f(A)]$是DQC1完备的。该条件适用于广泛函数类，包括指数函数、三角函数、对数函数及反函数类型。我们进一步证明：当A为稀疏矩阵时，该问题的经典查询复杂度随近似度呈指数增长（假设DQC1查询模型中k-Forrelation问题迹变体的下界猜想成立）。这些结果共同表明近似度是支配归一化迹估计复杂度的关键参数：它既刻画了量子复杂度（通过高效DQC1算法），又在条件假设下刻画了经典困难性，从而呈现出指数级量子-经典分离。我们的证明构建了统一框架，巧妙融合了电路到哈密顿量的构造、周期雅可比算子以及包括切比雪夫等振荡定理在内的多项式逼近理论工具。