Quantum simulation of a noisy classical nonlinear dynamics

TL;DR

The paper develops an end-to-end quantum algorithm for simulating noisy, strongly nonlinear classical dynamics with quadratic drift by mapping the stochastic differential equation to a backward Kolmogorov equation and encoding its second-quantized form on a system of quantum harmonic oscillators. A regularization scheme reduces the infinite-dimensional problem to a finite subspace, enabling efficient sparse Hamiltonian simulation, while a readout strategy expresses the target observable as an inner product with a readout state that accounts for initial-condition noise. The authors prove polynomial-time, poly-logarithmic-in-N scaling (up to an exponential penalty in initialization error) and establish BQP-completeness for the problem, underscoring a quantum advantage for this class of nonlinear, dissipative, and noisy dynamics. They also illustrate applicability to fluid dynamics by benchmarking on a 2D Navier–Stokes vortex flow, highlighting potential impact on turbulent system simulations once fault-tolerant quantum hardware becomes available.

Abstract

We present an end-to-end quantum algorithm for simulating nonlinear dynamics described by a system of stochastic dissipative differential equations with a quadratic nonlinearity. The stochastic part of the system is modeled by a Gaussian noise in the equation of motion and in the initial conditions. Our algorithm can approximate the expected value of any correlation function that depends on $O(1)$ variables with rigorous bounds on the approximation error. The runtime scales polynomially with $\log{N}$, $t$, $J$, and $λ_1^{-1}$, where $N$ is the total number of variables, $t$ is the evolution time, $J$ is the nonlinearity strength, and $λ_1$ is the smallest dissipation rate. However, the runtime scales exponentially with a parameter quantifying inverse relative error in the initial conditions. To the best of our knowledge, this is the first rigorous quantum algorithm capable of simulating strongly nonlinear systems with $J\gg λ_1$ at the cost poly-logarithmic in $N$ and polynomial in $t$. The considered simulation problem is shown to be BQP-complete, providing a strong evidence for a quantum advantage. We benchmark the quantum algorithm via numerical experiments by simulating a vortex flow in the 2D Navier Stokes equation.

Figures (4)

• Figure 1: Cartoon of our simulation problem. We consider a stochastic differential equation (SDE) for a state vector $X(t)\in \mathbb{R}^N$ with the initial condition $X(0)=x+z$, where $x$ is the desired initial condition and $z$ is a Gaussian noise. Red lines represent solutions of the SDE corresponding to different realization of noise, including Wiener noise in the SDE and noise in the initial condition. The goal is to approximate the expected value of a given observable function $u_0\, : \, \mathbb{R}^N\to \mathbb{R}$ evaluated at the solution $X(t)$. The expected value is taken over noise realizations.
• Figure 2: Quantum simulation of a nonlinear SDE. The workflow is divided into initialization, time evolution, and measurement stages. The simulator is initialized in a state $|\psi(0)\rangle$ encoding the observable function $u_0$ into the Hilbert space of $N$ quantum harmonic oscillators. Time evolution of $|\psi(t)\rangle$ is governed by a "Kolmogorian" $K=-A+B+C$, where $A=\sum_{i=1}^N \lambda_i a_i^\dag a_i$ describes noise and dissipation while $B$ and $C$ are anti-hermitian operators simply related to the drift functions $f_i(x)$. The Hilbert space is truncated to a finite dimension based on a novel regularization procedure. Time evolution under the truncated Kolmogorian is simulated using sparse Hamiltonian simulation methods. The measurement estimates the inner product between the time evolved state $|\psi(t)\rangle$ and a readout state $|\psi_{out}(x)\rangle$ encoding the desired initial condition $x$ of the SDE. The measured inner product gives an $\epsilon$-approximation of the classical expected value.
• Figure 3: Comparison of the solutions of the nse obtained by our approach and compared against true analytical solution: a) initial state of the Taylor-Green vortex flow; b) the case with vanishing noise; c) the case with non-vanishing noise: ensemble of $N_{\mathrm{rel}}$ members at 3 spatial resolutions (shades of blue) is compared agaisnt our algorithm (KE) (orange, red) with different $N$s and 3rd order polynomials; d) example of a realization of the velocity field (left) and the ensemble mean velocity field (right); e) realization of the vorticity and ensemble mean vorticity field.
• Figure 4: Solution of an incompressible 2D nse with $\nu=0.01$ with initial condition described by equation \ref{['eq:tg_vortex']} and perturbued with cylindrical Wiener noise with $\sigma^2=1$. The fields respectively represent $x$- and $y$-components of velocity and resulting vorticity.

Theorems & Definitions (42)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Corollary 1: Commutator bound
• proof
• Lemma 4: Order-$p$ commutator bound
• Lemma 5