Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-03-12 DOI:10.1137/22m1484298

Daan Camps, Lin Lin, Roel Van Beeumen, Chao Yang

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引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 801-827, March 2024.
Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [math] in a larger unitary transformation [math] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [math], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

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某些稀疏矩阵块编码的显式量子电路

SIAM 矩阵分析与应用期刊》，第 45 卷，第 1 期，第 801-827 页，2024 年 3 月。摘要。利用最近开发的量子线性代数算法和量子特征值/奇异值变换，可以在量子计算机上解决许多标准线性代数问题。分块编码将适当缩放的相关矩阵[数学]嵌入一个更大的单元变换[数学]中，该单元变换可分解为较简单单元的乘积，并在量子计算机上高效实现。虽然与最佳经典算法相比，量子算法在求解线性代数问题时有可能实现指数级的加速，但这种效率的提高最终取决于我们是否有能力为[数学]的分块编码构建高效的量子电路，而这在一般情况下是很困难的，即使对于结构良好的稀疏矩阵也并非易事。在本文中，我们举了几个例子，说明如何为一些结构良好的稀疏矩阵明确构建高效量子电路，并讨论了在这些构建中使用的一些策略。我们还提供了这些量子电路在 MATLAB 中的实现。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.