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 \(\mu _{t}\) of the solution \(X_{t}\) —appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on \(\mu _{t}\) through expectations of some functions \(\mathbb{E}[\varphi _{k}(X_{t})]\) , and the goal is to compute the expectation of a function \(\mathbb{E}[\phi (X_{T})]\) 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 \(\mathbb{E}[\phi (X_{T})]\) with accuracy ϵ , making \(O(1/\epsilon ^{1+2/p})\) queries to the quantum circuit for time evolution over one step, where \(p\in (1,2]\) 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 \(O(1/\epsilon ^{3})\) . 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.
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 (\mu {t}) of the solution (X{t}) —appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on (\mu {t}) through expectations of some functions (\mathbb{E}[\varphi {k}(X{t})]) , and the goal is to compute the expectation of a function (\mathbb{E}[\phi (X{T})]) 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 (\mathbb{E}[\phi (X_{T})]) with accuracy ϵ , making (O(1/\epsilon ^{1+2/p})) queries to the quantum circuit for time evolution over one step, where (p\in (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/\epsilon ^{3})) . 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.
Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.
Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of N-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in log(N) and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of N interacting quantum harmonic oscillators with cost polylogarithmic in N. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.
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.
Quantum Monte Carlo integration (QMCI) is a quantum algorithm to estimate expectations of random variables, with applications in various industrial fields such as financial derivative pricing. When QMCI is applied to expectations concerning a stochastic process X(t), e.g., an underlying asset price in derivative pricing, the quantum circuit UX(t) to generate the quantum state encoding the probability density of X(t) can have a large depth. With time discretized into N points, using state-preparation oracles for the transition probabilities of X(t), the state preparation for X(t) results in a depth of O(N), which may be problematic for large N. Moreover, if we estimate expectations concerning X(t) at N time points, the total query complexity scales on N as O(N2), which is worse than the O(N) complexity in the classical Monte Carlo method. In this paper, to improve this, we propose a method to divide UX(t) based on orthogonal-series-density estimation. This approach involves approximating the densities of X(t) at N time points with orthogonal series, where the coefficients are estimated as expectations of the orthogonal functions by QMCI. By using these approximated densities, we can estimate expectations concerning X(t) by QMCI without requiring deep circuits. Our error and complexity analysis shows that to obtain the approximated densities at N time points, our method achieves the circuit depth and total query complexity scaling as O(N) and O(N3/2), respectively.
Nonlinear stochastic differential equations (NSDEs) are a pillar of mathematical modeling for scientific and engineering applications. Accurate and efficient simulation of large-scale NSDEs is prohibitive on classical computers due to the large number of degrees of freedom, and it is challenging on quantum computers due to the linear and unitary nature of quantum mechanics. We develop a quantum algorithm to tackle nonlinear differential equations driven by the Ornstein-Uhlenbeck (OU) stochastic process. The query complexity of our algorithm scales logarithmically with the error tolerance and nearly quadratically with the simulation time. Our algorithmic framework comprises probabilistic Carleman linearization (PCL) to tackle nonlinearity coupled with stochasticity, and stochastic linear combination of Hamiltonian simulations (SLCHS) to simulate stochastic non-unitary dynamics. We obtain probabilistic exponential convergence for the Carleman linearization of Liu et al. [1], provided the NSDE is stable and reaches a steady state. We extend deterministic LCHS to stochastic linear differential equations, retaining near-optimal parameter scaling from An et al. [2] except for the nearly quadratic time scaling. This is achieved by using Monte Carlo integration for time discretization of both the stochastic inhomogeneous term in LCHS and the truncated Dyson series for each Hamiltonian simulation.
Quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), provide quadratic speed-up compared with classical counterparts, and are therefore widely investigated, along with applications to industries such as finance. One typical feature of problems in finance is high dimensionality, or, in other words, the necessity to create numerous random numbers (RNs) to calculate the integrand. In such a situation, the original implementation of QMCI, in which states encoding the probability distributions of the RNs are generated on different quantum registers, requires a large number of qubits. In this poster, we propose an implementation of QMCI, in which pseudo-random numbers (PRNs) are sequentially generated on one register, and therefore the qubit number is tremendously reduced. Moreover, we show that, when the integrand has the form such that contributions from different RNs are separable into different terms, we can also achieve time complexity reduction with respect to the number of dimensions by combining the nested QAE and use of PRNs.