In this paper, we present a quantum algorithm for approximating multivariate traces, i.e. the traces of matrix products. Our research is motivated by the extensive utility of multivariate traces in elucidating spectral characteristics of matrices, as well as by recent advancements in leveraging quantum computing for faster numerical linear algebra. Central to our approach is a direct translation of a multivariate trace formula into a quantum circuit, achieved through a sequence of low-level circuit construction operations. To facilitate this translation, we introduce \emph{quantum Matrix States Linear Algebra} (qMSLA), a framework tailored for the efficient generation of state preparation circuits via primitive matrix algebra operations. Our algorithm relies on sets of state preparation circuits for input matrices as its primary inputs and yields two state preparation circuits encoding the multivariate trace as output. These circuits are constructed utilizing qMSLA operations, which enact the aforementioned multivariate trace formula. We emphasize that our algorithm's inputs consist solely of state preparation circuits, eschewing harder to synthesize constructs such as Block Encodings. Furthermore, our approach operates independently of the availability of specialized hardware like QRAM, underscoring its versatility and practicality.

翻译：本文提出一种用于近似多元迹（即矩阵乘积的迹）的量子算法。本研究的动机源于多元迹在揭示矩阵谱特性方面的广泛应用价值，以及近期利用量子计算加速数值线性代数的发展趋势。我们方法的核心在于通过一系列底层电路构建操作，将多元迹公式直接转化为量子电路。为实现这一转化，我们引入量子矩阵态线性代数框架，该框架专为通过基本矩阵代数运算高效生成态制备电路而设计。我们的算法以输入矩阵的态制备电路集合为主要输入，最终生成两个编码多元迹的态制备电路作为输出。这些电路通过执行前述多元迹公式的qMSLA操作构建而成。值得强调的是，该算法的输入仅包含态制备电路，无需采用块编码等合成难度更高的结构。此外，我们的方法不依赖于QRAM等专用硬件的可用性，充分彰显其通用性与实用性。