Quantum equation simulates fluid flow with Schrödinger insights

Luca Cappelli of Fondazione Istituto Italiano di Tecnologia and colleagues have developed a new equation combining the Schrödinger equation with the Navier-Stokes equations, published August 12, 2026 in Quantum Science and Technology. This Schrödinger-Navier-Stokes equation aims to simulate fluid flow using principles from quantum mechanics, potentially enabling simulations of classical fluids on quantum computers. The work revisits a 1985 formulation, assessing its viability for quantum implementation through Carleman linearization. Researchers detail their findings in DOI 10.1088/2058-9565/ae9339.

Schrödinger-Navier-Stokes Equation for Fluid Simulation

A formulation of fluid dynamics from 1985, largely unnoticed until now, forms the basis for a new quantum simulation approach detailed in recent research. The team’s work addresses a longstanding challenge: accurately simulating classical fluids, with their inherent nonlinearities and dissipation, using the principles of quantum mechanics. The core of this advancement lies in adapting the SNS equation for Carleman linearization, a technique used to transform nonlinear systems into linear ones suitable for quantum computation.

However, the researchers discovered that standard application of Carleman linearization was obstructed by non-polynomial terms within the SNS equation related to dissipation and quantum pressure. To overcome this, they reformulated the dynamics as a Navier-Stokes-Hamilton-Jacobi (NSHJ) system, effectively enabling a quantum algorithm. This algorithm, termed Carleman-Hamilton-Jacobi (CHJ), incorporates a tensor-network representation that substantially reduces the memory requirements of its classical emulation.

To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the full NSE, including pressure, dissipation, and vorticity. The researchers tested the CHJ dynamics using classical computer simulations, analyzing its convergence and accuracy with Kolmogorov-like flows at moderate Reynolds numbers. The study highlights the importance of the Madelung formulation, a method of decomposing the wavefunction into probability density and a phase field, as a foundation for this hydrodynamic wave approach.

As the original source states, “devising an effective quantum algorithm for evolving this quantum system remains a valuable open problem,” a challenge Cappelli and his team appear to have directly addressed. Cappelli, whose correspondence details are listed with an ORCID ID of 0009-0009-1169-8380, emphasizes the mathematical simplicity and physical transparency of this approach compared to alternative methods involving spinorial wavefunctions.

The research, published in Quantum Science and Technology, details how the NSHJ system, derived from the SNS equation, allows for effective Carleman linearization and subsequent quantum implementation. The data cannot be made publicly available upon publication due to legal restrictions preventing unrestricted public distribution. The data that support the findings of this study are available upon reasonable request from the authors.

Dietrich & Vautherin’s 1985 SNS Formulation Revisited

The foundations for simulating fluid dynamics with quantum mechanics were laid remarkably early, with a 1985 formulation by Dietrich and Vautherin that received limited attention. This prior work, detailed in a paper titled “Sur l’équivalence entre des types particuliers des équations de Navier–Stokes et de Schrödinger non linéaire,” provides a mathematical framework for bridging classical fluid behavior and quantum computation. Researchers at Fondazione Istituto Italiano di Tecnologia, led by Luca Cappelli, have now revisited this formulation, assessing its potential for implementation on quantum computers.

This reveals both the promise and the challenges inherent in this approach. A key obstacle to directly applying standard Carleman linearization, a technique used to handle nonlinear equations in quantum computing, lies in the non-polynomial nature of the dissipative and quantum-pressure terms within the original Schrödinger-Navier-Stokes (SNS) equation.

Carleman Linearization of Navier-Stokes-Hamilton-Jacobi System

This transformation allows for a more tractable approach to linearization, enabling the development of a quantum algorithm termed Carleman-Hamilton-Jacobi (CHJ). The team’s analysis reveals that the CHJ scheme, while promising, requires careful consideration of parameters like Carleman truncation order and Reynolds number to achieve accurate and convergent results for complex fluid flows.

They note that previous attempts to integrate these elements into a quantum framework often relied on classical algorithms for handling dissipation, leaving a critical component unresolved. The data that support the findings of this study are available upon reasonable request from the authors, though legal restrictions prevent unrestricted public distribution.

CHJ Quantum Algorithm and Tensor-Network Representation

The work, identified by DOI 10. This transformation allows for effective Carleman linearization, embedding the system into a higher-dimensional linear one suitable for quantum processing. To further reduce the memory requirements, the researchers incorporated a tensor-network representation into the algorithm. This classical emulation served as an analysis of its convergence and accuracy, validating the method before potential implementation on actual quantum hardware. Cappelli led the effort to bridge the gap between quantum mechanics and classical fluid dynamics, potentially paving the way for more efficient and accurate simulations in the future.

Classical Emulation of CHJ Dynamics for Kolmogorov Flows

While quantum computing often evokes images of manipulating qubits, a recent effort detailed in DOI 10. 1088/2058-9565/ae9339 focuses on leveraging quantum principles to simulate classical systems, specifically the notoriously complex behavior of fluids. The team’s approach, dubbed Carleman-Hamilton-Jacobi (CHJ), does not immediately translate to a functioning quantum computer; a crucial validation step involved emulating the CHJ dynamics on a classical computer. This emulation analyzed its convergence and accuracy when applied to Kolmogorov-like flows, turbulent flows characterized by a cascade of energy across different scales.

The researchers specifically examined these flows at moderate Reynolds numbers, a dimensionless quantity indicating the ratio of inertial to viscous forces, to assess the CHJ scheme’s performance under realistic conditions. The success of this emulation provides confidence that the CHJ dynamics can be accurately represented before attempting implementation on actual quantum hardware.

Addressing Nonlinearity and Dissipation in Fluid Algorithms

The 1985 work of Dietrich and Vautherin, largely unnoticed, now forms the foundation for a novel approach to simulating fluid dynamics on quantum computers, as detailed in a recent publication with digital object identifier DOI 10.1088/2058-9565/ae9339. To overcome these limitations, the team transformed the original SNS equation into a Navier-Stokes-Hamilton-Jacobi (NSHJ) system, allowing for effective Carleman linearization. Classical emulation of the CHJ dynamics on conventional computers served as an analysis of its convergence and accuracy.

Madelung Formulation: Density and Phase Decomposition

This approach, initially developed for quantum mechanics, provides a crucial link to classical fluid behavior by establishing a parallel with hydrodynamic equations, effectively recasting them within a quantum-like framework. Researchers are leveraging this connection to design quantum algorithms capable of simulating complex fluid flows, a task currently limited by the computational demands of classical methods. The core of this formulation lies in separating the wavefunction, allowing for a continuity equation describing probability density and a quantum Hamilton-Jacobi equation governing the phase evolution.

This density-phase representation is not limited to purely quantum systems; it also underpins Bohmian mechanics, where particles follow deterministic paths dictated by the gradient of the wavefunction’s phase. Investigations have expanded this framework into areas like open quantum systems and stochastic dynamics, demonstrating its versatility beyond traditional quantum mechanics.

Studies have even explored the physical role of the Bohm potential and the impact of entanglement on particle motion, further solidifying the utility of this representation. More recently, the Madelung formulation has found application in areas beyond quantum mechanics. Transitioning from the quantum Madelung fluid to the classical Navier-Stokes equations requires three key steps: substituting the Bohm quantum potential with classical pressure, incorporating dissipation, and accounting for vorticity. This earlier work, largely unnoticed, provides a mathematically simpler and more transparent approach than alternative methods, relying on a scalar wavefunction coupled to a rotational external field.

Transition to Navier-Stokes Equations: Pressure, Dissipation, Vorticity

Fondazione Istituto Italiano di Tecnologia researcher Luca Cappelli is central to a renewed examination of a fluid dynamics equation first proposed in 1985. This work builds upon a largely unnoticed contribution by Dietrich and Vautherin published decades prior. The team’s analysis focuses on overcoming obstacles to implementing this Schrödinger-Navier-Stokes (SNS) equation on quantum computers. Previous attempts successfully addressed the first two steps, but a complete solution remained elusive until this recent work. The tensor-network representation substantially reduces the memory requirements of its classical emulation.