Over a decade ago, it was demonstrated that quantum computing has the potential to revolutionize numerical linear algebra by enabling algorithms with complexity superior to what is classically achievable, e.g., the seminal HHL algorithm for solving linear systems. Efficient execution of such algorithms critically depends on representing inputs (matrices and vectors) as quantum circuits that encode or implement these inputs. For that task, two common circuit representations emerged in the literature: block encodings and state preparation circuits. In this paper, we systematically study encodings matrices in the form of block encodings and state preparation circuits. We examine methods for constructing these representations from matrices given in classical form, as well as quantum two-way conversions between circuit representations. Two key results we establish (among others) are: (a) a general method for efficiently constructing a block encoding of an arbitrary matrix given in classical form (entries stored in classical random access memory); and (b) low-overhead, bidirectional conversion algorithms between block encodings and state preparation circuits, showing that these models are essentially equivalent. From a technical perspective, two central components of our constructions are: (i) a special constant-depth multiplexer that simultaneously multiplexes all higher-order Pauli matrices of a given size, and (ii) an algorithm for performing a quantum conversion between a matrix's expansion in the standard basis and its expansion in the basis of higher-order Pauli matrices.

翻译：十多年前的研究表明，量子计算有望通过实现复杂度超越经典极限的算法（例如求解线性系统的开创性HHL算法）彻底革新数值线性代数领域。此类算法的高效执行关键取决于将输入（矩阵与向量）表示为能编码或实现这些输入的量子电路。为此，文献中形成了两种主流的电路表示形式：块编码与态制备电路。本文系统研究了以块编码和态制备电路形式实现的矩阵编码方法。我们探讨了从经典形式给出的矩阵构建这些表示的方法，以及不同电路表示之间的量子双向转换技术。我们建立的两个关键成果（及其他成果）包括：（a）为经典形式给出的任意矩阵（条目存储于经典随机存取存储器）高效构建块编码的通用方法；（b）块编码与态制备电路之间的低开销双向转换算法，证明这两种模型本质上是等价的。从技术视角看，我们构建方案的两个核心组件是：（i）能同时复用给定尺寸所有高阶泡利矩阵的特殊常数深度多路复用器；（ii）实现矩阵在标准基展开式与高阶泡利矩阵基展开式之间量子转换的算法。