Quantum simulation of a noisy classical nonlinear dynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914072843255808 |
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| author | Bravyi, Sergey Manson-Sawko, Robert Zayats, Mykhaylo Zhuk, Sergiy |
| author_facet | Bravyi, Sergey Manson-Sawko, Robert Zayats, Mykhaylo Zhuk, Sergiy |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06198 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum simulation of a noisy classical nonlinear dynamics Bravyi, Sergey Manson-Sawko, Robert Zayats, Mykhaylo Zhuk, Sergiy Quantum Physics Mathematical Physics 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. |
| title | Quantum simulation of a noisy classical nonlinear dynamics |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2507.06198 |