[Paper Review] An efficient quantum algorithm for simulating polynomial differential equations

Amit Surana, Abeynaya Gnanasekaran+1 more|arXiv (Cornell University)|Dec 21, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a quantum algorithm for simulating nonlinear ordinary differential equations (ODEs) with polynomial vector fields of arbitrary finite degree $k$, extending prior work on quadratic systems. By mapping $k$-th degree ODEs to higher-dimensional quadratic systems and applying Carleman linearization followed by truncation and quantum linear system solving via forward Euler and QLSA, the algorithm achieves polynomial runtime scaling in $k$, $T$, and $\log n$, offering potential exponential speedups under specific conditions.

ABSTRACT

We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse $n$-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with $k$-th degree polynomial vector fields for arbitrary (finite) values of $k$. The steps involve: 1) mapping the $k$-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the $k$-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on $k$ and demonstrate the framework on an example.

Motivation & Objective

• To extend quantum algorithms for simulating nonlinear ODEs beyond quadratic vector fields to arbitrary finite-degree polynomial systems.
• To address the computational challenges of high dimensionality, stiffness, and nonlinearities in classical ODE/PDE solvers using quantum advantage.
• To maintain polynomial runtime scaling in system size $n$, evolution time $T$, and degree $k$ under prescribed conditions.
• To demonstrate that quantum speedup remains feasible for higher-degree polynomial ODEs through efficient embedding and truncation strategies.

Proposed method

• Mapping $k$-th degree polynomial ODEs into a higher-dimensional system of quadratic polynomial ODEs to enable application of Carleman linearization.
• Applying Carleman linearization to transform the quadratic ODE system into an infinite-dimensional linear ODE system.
• Truncating the infinite-dimensional system at a finite level $N$ to obtain a finite-dimensional approximation.
• Discretizing the truncated linear ODE system using the forward Euler method for numerical integration.
• Solving the resulting linear system using the quantum linear system algorithm (QLSA) to prepare a quantum state proportional to the solution.
• Optimizing the process by applying Carleman linearization directly to the $k$-th degree system, which can reduce computational cost compared to intermediate quadratic mapping.

Experimental results

Research questions

• RQ1 Can quantum algorithms simulate polynomial ODEs of arbitrary finite degree $k$ with polynomial runtime scaling in $k$?
• RQ2 Does the proposed method maintain quantum advantage over classical solvers for higher-degree nonlinearities?
• RQ3 How does the truncation level $N$ affect the accuracy and complexity of the solution in the Carleman linearization framework?
• RQ4 What is the impact of the solution decay factor $q = \|\mathbf{x}(0)\| / \|\mathbf{x}(T)\|$ on the overall query and gate complexity?
• RQ5 Can direct Carleman linearization of $k$-th degree systems outperform intermediate quadratic mapping in terms of computational efficiency?

Key findings

• The proposed algorithm achieves query and gate complexity of $\mathcal{O}\left(\frac{sT^{2}q}{\epsilon}\textsf{poly}(\log T, \log n, \log 1/\epsilon)\right)$, with polynomial scaling in $k$.
• The algorithm maintains polynomial runtime scaling in the evolution time $T$, overcoming the exponential dependence seen in prior approaches.
• Carleman linearization of $k$-th degree ODEs directly leads to a more computationally efficient route than intermediate quadratic mapping.
• The truncation level $N$ is chosen as $N = N'(k-1)$ to ensure all unique equations from the original system are captured in the linearized approximation.
• The complexity analysis confirms that the algorithm provides exponential speedup over classical methods under the condition $R_k < 1$, where $R_k$ characterizes nonlinearity relative to linear dissipation.
• Numerical demonstration confirms the feasibility and accuracy of the framework on a test example, validating the theoretical complexity bounds.