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Quantum Algorithms for Stochastic Differential Equations: Achieving Polynomial and Super-Polynomial Speedups for High-Dimensional Systems

Yalla Jnan Devi Satya Prasad

Published Jan 5, 2026• 0 citations
Publisher

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.

DOI
10.22541/au.176764501.12582212/v1

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Quantum Monte Carlo integration (QMCI), a quantum algorithm for calculating expectations that provides a quadratic speed-up compared to its classical counterpart, is now attracting increasing interest in the context of its industrial and scientific applications. In this paper, we propose the first application of QMCI to solving McKean-Vlasov stochastic differential equations (MVSDEs), a nonlinear class of SDEs whose drift and diffusion coefficients depend on the law μt\mu _{t}μt​ of the solution XtX_{t}Xt​ —appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on μt\mu _{t}μt​ through expectations of some functions E[φk(Xt)]\mathbb{E}[\varphi _{k}(X_{t})]E[φk​(Xt​)] , and the goal is to compute the expectation of a function E[ϕ(XT)]\mathbb{E}[\phi (X_{T})]E[ϕ(XT​)] at a terminal time T . We devise a quantum algorithm that leverages QMCI to compute these expectations, combined with a high-order time discretization method for SDEs and extrapolation of the expectations in time. The proposed algorithm estimates E[ϕ(XT)]\mathbb{E}[\phi (X_{T})]E[ϕ(XT​)] with accuracy ϵ , making O(1/ϵ1+2/p)O(1/\epsilon ^{1+2/p})O(1/ϵ1+2/p) queries to the quantum circuit for time evolution over one step, where p∈(1,2]p\in (1,2]p∈(1,2] is the weak order of the SDE discretization method. This demonstrates the speed-up over the well-known classical algorithm called the particle method with complexity of O(1/ϵ3)O(1/\epsilon ^{3})O(1/ϵ3) . We conduct numerical demonstrations of our quantum algorithm applied to examples of MVSDEs, with some parts emulated classically, and observe that the accuracy and complexity behave as expected.
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