Abstract Noisy linear problems have been studied in various science and engineering disciplines. A class of ‘hard’ noisy linear problems can be formulated as follows: Given a matrix A ^ and a vector b constructed using a finite set of samples, a hidden vector or structure involved in b is obtained by solving a noise-corrupted linear equation A ^ x ≈ b + η , where η is a noise vector that cannot be identified. For solving such a noisy linear problem, we consider a quantum algorithm based on a divide-and-conquer strategy, wherein a large core process is divided into smaller subprocesses. The algorithm appropriately reduces both the computational complexities and size of a quantum sample. More specifically, if a quantum computer can access a particular reduced form of the quantum samples, polynomial quantum-sample and time complexities are achieved in the main computation. The size of a quantum sample and its executing system can be reduced, e.g., from exponential to sub-exponential with respect to the problem length, which is better than other results we are aware. We analyse the noise model conditions for such a quantum advantage, and show when the divide-and-conquer strategy can be beneficial for quantum noisy linear problems.

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分治策略对噪声线性问题的量子可解性

摘要 噪声线性问题已在各个科学和工程学科中得到研究。一类“硬”噪声线性问题可以表述如下：给定一个矩阵 A ^ 和一个使用有限样本集构建的向量 b，b 中涉及的隐藏向量或结构是通过求解噪声破坏的线性方程 A ^ x ≈ b + η 获得的，其中 η 是无法识别的噪声向量。为了解决这样一个嘈杂的线性问题，我们考虑了一种基于分治策略的量子算法，其中一个大的核心进程被划分为更小的子进程。该算法适当地降低了量子样本的计算复杂性和大小。更具体地说，如果量子计算机可以访问特定简化形式的量子样本，则在主计算中实现多项式量子样本和时间复杂度。量子样本及其执行系统的大小可以减少，例如，就问题长度而言，从指数减少到次指数，这比我们知道的其他结果要好。我们分析了这种量子优势的噪声模型条件，并展示了分治策略何时可以有益于量子噪声线性问题。多项式量子样本和时间复杂度在主计算中实现。量子样本及其执行系统的大小可以减少，例如，就问题长度而言，从指数减少到次指数，这比我们知道的其他结果要好。我们分析了这种量子优势的噪声模型条件，并展示了分治策略何时可以有益于量子噪声线性问题。多项式量子样本和时间复杂度在主计算中实现。量子样本及其执行系统的大小可以减少，例如，就问题长度而言，从指数减少到次指数，这比我们知道的其他结果要好。我们分析了这种量子优势的噪声模型条件，并展示了分治策略何时可以有益于量子噪声线性问题。