Scholarlink

<p>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.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

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

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年8月18日

查看全文
  • Pedro C. S. Costa,
  • Philipp Schleich,
  • Mauro E. S. Morales,
  • Dominic W. Berry

摘要

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.