We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. For $N\times N$ Hermitian matrices, the space cost is $\log(N)+1$ qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size $O(N^2)$, when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as an algorithm for sampling properties of ground states of Hamiltonians. As a concrete application, we combine these two sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

翻译：我们提出了一类用于矩阵函数采样的随机量子算法，这类算法无需使用量子块编码或任何其他相干量子访问矩阵元素的方式。因此，我们对量子比特的使用纯粹是算法性的，无需额外量子比特用于量子数据结构。对于$N\times N$的厄米矩阵，空间成本为$\log(N)+1$个量子比特。根据矩阵结构，其门复杂度可媲美使用规模高达$O(N^2)$量子数据结构的先进方法（在考虑等效端到端问题的情况下）。在此框架内，我们提出了一种量子线性系统求解器，可实现对解向量性质的采样，以及一种哈密顿量基态性质采样的算法。作为具体应用，我们将这两个子程序相结合，提出了计算量子多体系统格林函数的方案。