通过高阶方法和重标度进一步改进非线性微分方程的量子算法

IF 8.3 1区 物理与天体物理 Q1 PHYSICS, APPLIED
Pedro C. S. Costa, Philipp Schleich, Mauro E. S. Morales, Dominic W. Berry
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引用次数: 0

摘要

大型非线性微分方程组的解在科学和工程中的许多应用中是必不可少的。我们提出了基于Carleman线性化技术的现有量子算法的三种改进。首先,我们使用高精度方法来求解线性化系统,该系统对误差产生对数依赖,对时间产生近线性依赖。其次,我们引入了一种可显着降低成本的重新缩放策略,否则该策略将随卡尔曼顺序呈指数级扩展,从而限制了pde的量子速度。第三,我们为卡尔曼线性化导出了更严格的误差界。我们将我们的结果应用于一类离散反应扩散方程,使用高阶有限差分进行空间分辨率。我们还表明,由于最大范数和2范数之间的不匹配,强制独立于离散化的稳定性准则可能与重新缩放相冲突。尽管如此,当离散点的数量受到限制时,高效的量子解决方案仍然是可能的,正如高阶方案所实现的那样。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling

Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling

The solution of large systems of nonlinear differential equations is essential for many applications in science and engineering. We present three improvements to existing quantum algorithms based on the Carleman linearisation technique. First, we use a high-precision method for solving the linearised system that yields logarithmic dependence on the error and near-linear dependence on time. Second, we introduce a rescaling strategy that significantly reduces the cost, which would otherwise scale exponentially with the Carleman order, thus limiting quantum speedups for PDEs. Third, we derive tighter error bounds for Carleman linearisation. We apply our results to a class of discretised reaction-diffusion equations using higher-order finite differences for spatial resolution. We also show that enforcing a stability criterion independent of the discretisation can conflict with rescaling due to the mismatch between the max-norm and the 2-norm. Nonetheless, efficient quantum solutions remain possible when the number of discretisation points is constrained, as enabled by higher-order schemes.

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来源期刊
npj Quantum Information
npj Quantum Information Computer Science-Computer Science (miscellaneous)
CiteScore
13.70
自引率
3.90%
发文量
130
审稿时长
29 weeks
期刊介绍: The scope of npj Quantum Information spans across all relevant disciplines, fields, approaches and levels and so considers outstanding work ranging from fundamental research to applications and technologies.
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