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. Our algorithms start from a classical data structure in which the matrix of interest is specified in the Pauli basis. 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，无需使用量子块编码或任何对矩阵元素的其他相干神谕访问。因此，我们对量子比特的使用纯粹是算法性的，且无需为量子数据结构提供额外量子比特。我们的算法从经典数据结构出发，其中目标矩阵以泡利基表示。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，门复杂度可与使用规模达$O(N^2)$的量子数据结构的最先进方法相当（当考虑等效端到端问题时）。在此框架内，我们提出了一种量子线性系统求解器，可对解向量性质进行采样，以及用于对哈密顿量的基态和吉布斯态性质进行采样的算法。作为具体应用，我们将这些子程序结合，提出了一种用于计算量子多体系统格林函数的方案。