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Quantum algorithm for solving McKean-Vlasov stochastic differential equations

Koichi Miyamoto

TL;DR

The paper addresses solving nonlinear McKean-Vlasov SDEs by introducing a quantum Monte Carlo integration framework that uses QMCI to estimate law-dependent expectations γ_k(t) and an extrapolated, high-order time discretization to control the time-step count. The core approach combines a stochastic Runge-Kutta discretization with linear extrapolation of γ_k(t), yielding an overall query complexity of O(1/ε^{1+2/p}) for estimating E[φ(X_T)], which surpasses the classical particle method cost of O(1/ε^3). Theoretical results accompany a numerical demonstration on two MVSDE examples, verifying both accuracy and the predicted cost scaling when some components are emulated classically. This work demonstrates a concrete quantum speed-up path for nonlinear stochastic dynamics and points to practical routes for quantum-accelerated derivative pricing and related applications.

Abstract

Quantum Monte Carlo integration, 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 $μ_t$ 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 $μ_t$ through expectations of some functions $\mathbb{E}[\varphi_k(X_t)]$, and the goal is to compute the expectation of a function $\mathbb{E}[φ(X_T)]$ 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 $\mathbb{E}[φ(X_T)]$ with accuracy $ε$, making $O(1/ε^{1+2/p})$ 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 $O(1/ε^3)$. We conduct a numerical demonstration of our quantum algorithm applied to an example of MVSDEs, with some parts emulated classically, and observe that the accuracy and complexity behave as expected.

Quantum algorithm for solving McKean-Vlasov stochastic differential equations

TL;DR

The paper addresses solving nonlinear McKean-Vlasov SDEs by introducing a quantum Monte Carlo integration framework that uses QMCI to estimate law-dependent expectations γ_k(t) and an extrapolated, high-order time discretization to control the time-step count. The core approach combines a stochastic Runge-Kutta discretization with linear extrapolation of γ_k(t), yielding an overall query complexity of O(1/ε^{1+2/p}) for estimating E[φ(X_T)], which surpasses the classical particle method cost of O(1/ε^3). Theoretical results accompany a numerical demonstration on two MVSDE examples, verifying both accuracy and the predicted cost scaling when some components are emulated classically. This work demonstrates a concrete quantum speed-up path for nonlinear stochastic dynamics and points to practical routes for quantum-accelerated derivative pricing and related applications.

Abstract

Quantum Monte Carlo integration, 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 of the solution -- appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on through expectations of some functions , and the goal is to compute the expectation of a function at a terminal time . 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 with accuracy , making queries to the quantum circuit for time evolution over one step, where 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 . We conduct a numerical demonstration of our quantum algorithm applied to an example of MVSDEs, with some parts emulated classically, and observe that the accuracy and complexity behave as expected.

Paper Structure

This paper contains 21 sections, 3 theorems, 81 equations, 9 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notation
  4. Problem
  5. Time discretization of SDEs
  6. Particle method for MVSDEs
  7. Extrapolation of a function
  8. Quantum building-blocks
  9. Representing numbers on qubits
  10. Quantum arithmetic circuits
  11. Loading probability distributions into quantum states
  12. Quantum Monte Carlo integration
  13. Proposed quantum algorithm
  14. Idea
  15. Algorithm
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\epsilon\in\mathbb{R}_+$ and $\eta\in(0,1)$. Suppose that $X$ is a $\mathbb{R}^d$-valued random variable with density $q$. Suppose that for a function $f:\mathbb{R}^d\rightarrow \mathbb{R}$, $\|f(X)\|_{2}^2=\int f^2(x) q(x) dx\le U <\infty$ exists with a known upper bound $U>1$. Suppose that fo times.

Figures (9)

  • Figure 1: The quantum circuit $U_{\tilde{X}_{i+1}}$ to generate the state that encodes the distribution of $\tilde{X}_{i+1}$.
  • Figure 2: The blue curve represents the root mean square deviation of the outputs $\hat{E}$ from the true value of $\mathbb{E}[X_T]$ in 10 runs of Algorithm \ref{['alg:mainSim']} for the Shimizu-Yamada model, where the parameters are set as Eqs. \ref{['eq:ProbParam']} and \ref{['eq:AlgoParam']} with varying $\varepsilon$. The red line is the slope-1 line fitted to the blue curve using least squares in the log-log space.
  • Figure 3: $\mathbb{E}[X_{t}]$ estimated by Algorithm \ref{['alg:mainSim']} with $\varepsilon=1/12$ (blue) and its exact value (red) for the Shimizu-Yamada model.
  • Figure 4: The blue curve represents the total number of queries to $U_{F^h_{a,b,t}}$ in Algorithm \ref{['alg:mainSim']} for the Shimizu-Yamada model versus the realized error in the algorithm, which is plotted in FIG. \ref{['fig:RMSE']}. Here, we assume that $\textup{QMCIML}$ in it were run as a quantum algorithm that queries $U_{F^h_{a,b,t}}$. The red one represents the function $y=a/x^2$ with $a$ tuned via the least-squares fitting to the blue curve in the log-log space.
  • Figure 5: The blue curve represents the root mean square deviation of the outputs $\hat{E}$ from the benchmark value of $\mathbb{E}[\sin(\theta_T)]$ in 10 runs of Algorithm \ref{['alg:mainSim']} for the KSS model, where the parameters are set as Eqs. \ref{['eq:ProbParamK']} and \ref{['eq:AlgoParam']} (except $u_i$ and $l_i$ set to $\pm1$) with varying $\varepsilon$. The benchmark is the Euler discretization-based particle method with $N=10^6$ and $h=T/256$. The red line is the slope-1 line fitted to the blue curve using least squares in the log-log space.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1: Lemma 2.4 in montanaro2015, modified
  • Theorem 2
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['th:main']}