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This preprint studies quantum algorithms for simulating and solving stochastic differential equations (SDEs) in high-dimensional settings, using approaches such as Hamiltonian simulation for drift, quantum random number generation for noise increments, and quantum amplitude estimation for expectations. For linear and semilinear SDEs with Lipschitz coefficients, it proves an error-to-runtime scaling of Õ(poly(d)polylog(1/ε)) for estimating quantities like option prices, expected hitting times, and moments, compared to classical O(poly(d)/ε²). For nonlinear SDEs, it proposes variational quantum schemes with parameterized circuits, providing error bounds and a trainability analysis aimed at avoiding barren plateaus, while also analyzing required resources (qubits, gate depth, and oracle calls) on example models including multi-asset Black–Scholes, Heston stochastic volatility, and Langevin dynamics. The paper’s limitation is that its strongest rigorous results apply to linear/semilinear Lipschitz SDEs and that the nonlinear case relies on variational methods whose performance depends on the provided trainability considerations. The paper does not explicitly discuss endometriosis or adenomyosis; it was included in the corpus via a keyword match in the upstream search index.
Abstract
Stochastic differential equations (SDEs) model noisy dynamical systems across quantitative finance, chemistry, physics, and climate science. Classical solvers-Euler-Maruyama schemes, Monte Carlo methods, and finite-difference PDE solvers for Fokker-Planck equations-suffer from exponential growth in computational cost with dimension and accuracy requirements. We present a family of quantum algorithms for simulating and solving SDEs with improved scaling. Our approach combines Hamiltonian simulation for drift terms, quantum random number generation for noise increments, and quantum amplitude estimation for computing expectations. For linear and semilinear SDEs with Lipschitz coefficients, we prove that relevant quantities (option prices, expected hitting times, moments) can be estimated to accuracy ε in time Õ(poly(d)polylog(1/ε)), versus classical O(poly(d)/ε 2). For nonlinear SDEs, we introduce variational quantum schemes leveraging parameterized circuits, with a priori error bounds and trainability analysis avoiding barren plateaus. We analyze multi-asset Black-Scholes models, Heston stochastic volatility, and Langevin dynamics for molecular coarse-graining, quantifying resource requirements (qubits, gate depth, oracle calls). Numerical simulations on small instances demonstrate polynomial-to-super-polynomial empirical speedups in accuracy for fixed runtime, establishing SDE solving as a promising quantum-accelerated task with practical applications.
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Quantum Algorithms for Stochastic Differential Equations: Achieving Polynomial and Super-Polynomial Speedups for High-Dimensional Systems | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 5 January 2026 V1 Latest version Share on Quantum Algorithms for Stochastic Differential Equations: Achieving Polynomial and Super-Polynomial Speedups for High-Dimensional Systems Author : Yalla Jnan Devi Satya Prasad 0009-0000-6343-3733 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176764501.12582212/v1 171 views 90 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Stochastic differential equations (SDEs) model noisy dynamical systems across quantitative finance, chemistry, physics, and climate science. Classical solvers-Euler-Maruyama schemes, Monte Carlo methods, and finite-difference PDE solvers for Fokker-Planck equations-suffer from exponential growth in computational cost with dimension and accuracy requirements. We present a family of quantum algorithms for simulating and solving SDEs with improved scaling. Our approach combines Hamiltonian simulation for drift terms, quantum random number generation for noise increments, and quantum amplitude estimation for computing expectations. For linear and semilinear SDEs with Lipschitz coefficients, we prove that relevant quantities (option prices, expected hitting times, moments) can be estimated to accuracy ε in time Õ(poly(d)polylog(1/ε)), versus classical O(poly(d)/ε 2). For nonlinear SDEs, we introduce variational quantum schemes leveraging parameterized circuits, with a priori error bounds and trainability analysis avoiding barren plateaus. We analyze multi-asset Black-Scholes models, Heston stochastic volatility, and Langevin dynamics for molecular coarse-graining, quantifying resource requirements (qubits, gate depth, oracle calls). Numerical simulations on small instances demonstrate polynomial-to-super-polynomial empirical speedups in accuracy for fixed runtime, establishing SDE solving as a promising quantum-accelerated task with practical applications. Supplementary Material File (quantum sde algorithms.pdf) Download 246.45 KB Information & Authors Information Version history V1 Version 1 05 January 2026 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords differential equations hybrid algorithms quality quantum algorithms quantum computing quantum machine learning Authors Affiliations Yalla Jnan Devi Satya Prasad 0009-0000-6343-3733 [email protected] IQ Leap Pvt Ltd View all articles by this author Metrics & Citations Metrics Article Usage 171 views 90 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Yalla Jnan Devi Satya Prasad. Quantum Algorithms for Stochastic Differential Equations: Achieving Polynomial and Super-Polynomial Speedups for High-Dimensional Systems. Authorea . 05 January 2026. DOI: https://doi.org/10.22541/au.176764501.12582212/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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