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Efficient Quantum Simulation for Nonlinear Stochastic Differential Equations

Xiangyu Li, Ahmet Burak Catli, Ho Kiat Lim, Matthew Pocrnic, Dong An, Jin-Peng Liu, Nathan Wiebe

Abstract

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.

Efficient Quantum Simulation for Nonlinear Stochastic Differential Equations

Abstract

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.
Paper Structure (25 sections, 47 theorems, 479 equations)

This paper contains 25 sections, 47 theorems, 479 equations.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Main results
  4. Organization
  5. Review of Carleman linearization for nonlinear differential equations
  6. Review of stochastic differential equations
  7. Probabilistic bounds for the norm of a multivariate Ornstein--Uhlenbeck process
  8. Probabilistic Carleman linearization for NSDEs
  9. Carleman linearization for steady-state NSDEs
  10. Probabilistic Carleman linearization error bounds
  11. Stochastic linear combination of Hamiltonian simulations
  12. LCHS for linear SDEs
  13. Stochastic inhomogeneous term
  14. Truncated Dyson series simulation
  15. Quantum implementation and complexity for NSDEs
  16. ...and 10 more sections

Key Result

Lemma 4

[lemma]lemma:RNt For the residual term $\mathbf{R}_N(t)$ in the truncated Carleman linearization, the following bound holds:

Theorems & Definitions (109)

  • Definition 2
  • Definition 3: Carleman linearization
  • Lemma 4: Bound on the Carleman Residual Term
  • proof
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 99 more