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 algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

翻译：我们提出了一类随机量子算法，用于从矩阵函数中采样的任务，无需使用量子块编码或任何其他对矩阵元素的相干神谕访问。因此，我们对量子比特的使用纯粹是算法性的，无需额外的量子比特用于量子数据结构。对于$N\times N$埃尔米特矩阵，空间开销为$\log(N)+1$个量子比特，并且根据矩阵结构，门复杂度可与最先进的方法相媲美——这些方法在使用规模高达$O(N^2)$的量子数据结构时，在等效的端到端问题下具有竞争力。在我们的框架内，我们提出了一种量子线性系统求解器，允许对解向量的属性进行采样，以及用于对哈密顿量的基态和吉布斯态属性进行采样的算法。作为一个具体应用，我们将这些子程序结合起来，提出了一种计算量子多体系统格林函数的方案。