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<p>Quantum computers are known for their potential to achieve up-to-exponential speedup compared to classical computers for certain problems. To exploit the advantages of quantum computers, we propose quantum algorithms for linear stochastic differential equations, utilizing the Schrödingerisation method for the corresponding approximate equation by treating the noise term as a (discrete-in-time) forcing term. Our algorithms are applicable to stochastic differential equations with both Gaussian noise and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2970_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable Lévy noise. The gate complexity of our algorithms exhibits an <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2970_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {O}}(d\log (Nd))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>log</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> dependence on the dimensions <i>d</i> and sample sizes <i>N</i>, where its corresponding classical counterpart requires nearly exponentially larger complexity in scenarios involving large sample sizes. In the Gaussian noise case, we show the strong convergence of first order in the mean square norm for the approximate equations. The algorithms are numerically verified for the Ornstein–Uhlenbeck processes, geometric Brownian motions, and one-dimensional Lévy flights.</p>

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Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach

  • 期刊论文

  • 发表日期：

    2025年6月29日

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  • Shi Jin,
  • Nana Liu,
  • Wei Wei

摘要

Quantum computers are known for their potential to achieve up-to-exponential speedup compared to classical computers for certain problems. To exploit the advantages of quantum computers, we propose quantum algorithms for linear stochastic differential equations, utilizing the Schrödingerisation method for the corresponding approximate equation by treating the noise term as a (discrete-in-time) forcing term. Our algorithms are applicable to stochastic differential equations with both Gaussian noise and \(\alpha \) α -stable Lévy noise. The gate complexity of our algorithms exhibits an \({\mathcal {O}}(d\log (Nd))\) O ( d log ( N d ) ) dependence on the dimensions d and sample sizes N, where its corresponding classical counterpart requires nearly exponentially larger complexity in scenarios involving large sample sizes. In the Gaussian noise case, we show the strong convergence of first order in the mean square norm for the approximate equations. The algorithms are numerically verified for the Ornstein–Uhlenbeck processes, geometric Brownian motions, and one-dimensional Lévy flights.