⚛️ quantum physics

Hamiltonian simulation for nonlinear partial differential equation by Schrödingerization

This paper proposes a method called Carleman linearization + Schrödingerization (CLS) that combines Carleman linearization and warped phase transformation to convert nonlinear partial differential equations into the Schrödinger equation, thereby enabling their efficient solution via Hamiltonian simulation on quantum computers.

Original authors: Shoya Sasaki, Katsuhiro Endo, Mayu Muramatsu

Published 2026-08-11

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Shoya Sasaki, Katsuhiro Endo, Mayu Muramatsu

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a drop of ink spreads in a glass of water, or how a fire burns through a forest. These aren't just random events; they follow strict mathematical rules called "partial differential equations" (PDEs). Think of these equations as the ultimate instruction manual for how things change over time and space. For decades, scientists have used powerful supercomputers to solve these manuals, but when the systems get huge—like modeling the entire atmosphere or the flow of blood through every vessel in the body—the math becomes so heavy that even the best computers get stuck.

Enter the world of quantum computing. You might have heard that quantum computers are like magical dice that can roll every possible outcome at once. Because of this superpower, they promise to solve these massive math puzzles much faster than regular computers. One specific trick they use is called "Hamiltonian simulation." Imagine a quantum computer as a giant, complex music box. If you can tune the gears (the "Hamiltonian") just right, the music box will naturally play the exact song of how a physical system evolves, without needing to calculate every single note step-by-step. This works beautifully for simple, linear systems—like a perfect spring bouncing up and down. But the real world is messy. Most interesting things, like turbulent water or chemical reactions, are "nonlinear," meaning their behavior is chaotic and doesn't fit the neat, straight lines that quantum music boxes usually play. The big question has been: Can we teach a quantum computer to play the messy, nonlinear songs of the real world?

This paper by Shoya Sasaki, Katsuhiro Endo, and Mayu Muramatsu says, "Yes, but we need a translator." The authors propose a new method they call CLS (Carleman Linearization + Schrödingerization). Think of it as a two-step translation process to help a quantum computer understand a chaotic, nonlinear story.

First, they use a technique called Carleman Linearization. Imagine you have a story written in a complex, slang-heavy language (the nonlinear equation) that a strict librarian (the quantum computer) refuses to read. The authors take that story and rewrite it into a massive, infinite library of simple, straight-line sentences (a linear system). They do this by expanding the story into higher dimensions, essentially adding more and more characters to the plot until the complex interactions look like simple, straight lines. However, there's a catch: this new "linear" story is a bit "leaky"—mathematically speaking, it's a "dissipative system," meaning it loses energy over time, like a spinning top that eventually stops.

Second, they use a technique called Schrödingerization (specifically using Warped Phase Transformation). Quantum computers are picky; they only like stories that are perfectly balanced and never lose energy (conservative systems). To fix the "leaky" story from the first step, the authors introduce a brand-new, invisible dimension called $p$ . Imagine adding a new axis to your graph, like a hidden slider that controls the volume. By stretching the story into this new dimension, they can transform the "leaky" system into a perfectly balanced, energy-conserving one. This transformed story is now a "Schrödinger equation," which is exactly the kind of music a quantum computer knows how to play.

The authors tested this method on a classic nonlinear problem: the reaction–diffusion equation. This equation describes how two things happen at once: a substance spreading out (diffusion) and reacting with itself (like a chemical changing color). They simulated this on a classical computer to see if their CLS method worked. The results were promising: the CLS method produced a solution that looked almost identical to the standard, trusted methods used today.

However, the paper also reveals where the magic loses a little bit of its sparkle. When they checked the accuracy, they found that the CLS method is "first-order accurate" regarding the new hidden dimension ( $p$ ) and the complexity of the linearization (the "Carleman order"). This means that to get a super-precise answer, you have to make your grid of points in that hidden dimension very fine, which requires a lot of computing power. They also noted that the "leakage" in the hidden dimension can cause errors to creep in over long simulations, much like a small wave in a pool eventually hitting the shore and splashing back in.

In short, this paper doesn't claim to have solved all nonlinear problems instantly. Instead, it suggests a clever new roadmap. It shows that by combining a way to straighten out the curves (Carleman) with a way to hide the leaks in a new dimension (Schrödingerization), we can indeed guide quantum computers to simulate the messy, nonlinear world. It's a proof of concept that opens the door, even if we still have some work to do to make the journey perfectly smooth.

Technical Summary: Hamiltonian Simulation for Nonlinear PDEs via Schrödingerization

Problem Statement
Hamiltonian simulation is a fundamental quantum algorithm capable of efficiently solving the governing equations of large-scale classical systems, with rigorous demonstrations of exponential speedup for linear systems such as coupled harmonic oscillators. However, a significant limitation exists: the Schrödinger equation, which underpins Hamiltonian simulation, is inherently linear, whereas many real-world physical phenomena (e.g., turbulence, large material deformations, reaction–diffusion systems) are governed by nonlinear partial differential equations (PDEs).

Performing Hamiltonian simulations on nonlinear PDEs presents two primary challenges:

Nonlinearity: The target nonlinear PDE must be transformed into a linear equation, as quantum Hamiltonian evolution is linear.
Dissipation: Linearized equations often result in dissipative systems (non-unitary time evolution), whereas Hamiltonian simulation requires conservative systems (unitary time evolution).

Existing methods for linearization (e.g., Koopman von Neumann, Carleman linearization) and handling dissipation (e.g., unitary dilation, Linear Combination of Hamiltonian Simulation) have limitations, such as probabilistic success rates or specific constraints on basis functions.

Methodology: Carleman Linearization + Schrödingerization (CLS)
The authors propose a framework named Carleman Linearization + Schrödingerization (CLS) to address these challenges. The method integrates two distinct techniques to transform a nonlinear PDE into a form amenable to Hamiltonian simulation:

Carleman Linearization (CL):
- The nonlinear PDE (specifically a nonlinear reaction–diffusion equation) is first transformed into an infinite-dimensional system of linear ordinary differential equations (ODEs).
- This is achieved by extending the state variables into an infinite-dimensional space using Kronecker products of the original state vector.
- Since infinite dimensions are computationally intractable, the system is truncated at a finite order $K$ , resulting in a finite-dimensional linear ODE system.
- The authors note that this truncated linear system is generally a dissipative system.
Schrödingerization via Warped Phase Transformation (WPT):
- To enable Hamiltonian simulation, the dissipative linear ODE is mapped to a conservative Schrödinger equation.
- The core of this step is the Warped Phase Transformation (WPT), which introduces an auxiliary variable $p$ (independent of spatial dimensions).
- The transformation is defined as $\psi(t, p) = e^{-p}\Phi(t)$ , where $\Phi(t)$ is the state vector of the linearized system.
- By decomposing the system matrix $A$ into Hermitian ( $H_1$ ) and skew-Hermitian ( $iH_2$ ) parts, the evolution equation is rewritten to include a spatial derivative term in the $p$ -direction ( $-\partial/\partial p$ ).
- This results in a new system where the time evolution operator is unitary (skew-Hermitian), satisfying the requirements for Hamiltonian simulation.
- The $p$ -direction is discretized using a first-order upwind difference method to handle the advection of the solution in the negative $p$ -direction.

Key Contributions

Framework Development: The paper introduces the CLS framework, which successfully bridges the gap between nonlinear PDEs and quantum Hamiltonian simulation by combining Carleman linearization and Warped Phase Transformation.
Application to Nonlinear PDEs: The method is applied to the nonlinear reaction–diffusion equation, demonstrating a pathway to simulate nonlinear dynamics using quantum algorithms without relying on probabilistic post-selection methods (unlike unitary dilation or LCHS).
Error Analysis: The study provides a detailed error analysis of the CLS method, breaking down accuracy contributions from the truncation order, spatial discretization, and the auxiliary variable discretization.

Results
The authors evaluated the CLS method using classical numerical simulations to verify its utility and accuracy against Finite Difference Method (FDM) solutions and standard Carleman linearization (CL) solutions.

Qualitative Agreement: The time evolution of the solution obtained via CLS qualitatively coincides with solutions obtained via FDM and standard CL.
Accuracy Characteristics:
- Truncation Order ( $K$ ): The relative error of the CL component scales inversely with the truncation order, indicating first-order accuracy with respect to $K$ (i.e., error decreases linearly as $K$ increases).
- Spatial Discretization ( $x$ ): The error in the spatial direction scales with the square of the grid spacing, demonstrating second-order accuracy, consistent with the use of the second-order central difference method.
- Auxiliary Discretization ( $p$ ): The error in the $p$ -direction scales linearly with the grid spacing, showing first-order accuracy, consistent with the first-order upwind difference method.
Error Sources: The analysis indicates that the error introduced by the WPT process (specifically the discretization in the $p$ -direction and advection effects) dominates the total error of the CLS method compared to the error introduced by CL alone.
Advection Effects: The study observed that extending the computational domain to $p < 0$ (to accommodate the initial condition $e^{-|p|}$ ) can introduce errors in long-term simulations if the advected wave from the $p < 0$ region interacts with the $p \ge 0$ region, despite periodic boundary conditions.

Significance and Claims
The paper claims that the proposed CLS method extends the framework of time evolution simulation in quantum computing. By successfully transforming a nonlinear PDE into a Schrödinger equation, the study demonstrates that Hamiltonian simulation is applicable to nonlinear PDEs.

The authors maintain a modest stance regarding the current state of the method:

The primary objective was to investigate the utility and characteristics of CLS through classical simulations, not to execute the algorithm on a quantum computer.
The study confirms that the computational accuracy aligns with theoretical predictions based on the chosen discretization schemes.
While the method enables the analysis of original nonlinear equations via transformed Schrödinger equations, the paper does not claim an exponential speedup for the nonlinear case itself, noting that rigorous speedup has only been demonstrated for linear systems (coupled harmonic oscillators) to date. The work serves as a proof-of-concept for the applicability of Hamiltonian simulation to nonlinear governing equations.

Technical Summary: Hamiltonian Simulation for Nonlinear PDEs via Schrödingerization

More like this