⚛️ quantum physics

Simulation of a Battery Cell on Quantum Computers: Reactions & Transport

This paper presents a scalable hybrid quantum-classical algorithm for simulating non-linear partial differential equations in electrochemical systems, demonstrating its application by performing the first quantum simulation of a battery cell using the Single Particle Model with electrolyte (SPMe).

Original authors: Albert J. Pool, Michael Schelling, Birger Horstmann

Published 2026-09-24

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CC BY 4.0

Original authors: Albert J. Pool, Michael Schelling, Birger Horstmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

Every modern battery relies on a hidden, invisible dance of atoms and ions moving through porous materials, a process that scientists have long tried to predict with powerful computers. To understand how a battery charges or discharges, researchers must solve complex mathematical descriptions of how substances spread and react, known as partial differential equations. These equations are the foundation of engineering, allowing us to model everything from weather patterns to the flow of electricity inside a cell. However, as scientists demand more detailed and accurate predictions, the sheer volume of data required to run these simulations on traditional computers has begun to hit a wall. The physical limits of current computer chips mean that simulating these intricate chemical systems with high precision is becoming increasingly difficult, prompting a search for a new kind of machine capable of handling this complexity.

A team of researchers at the German Aerospace Center and Ulm University has taken a significant step toward this new frontier by running a simulation of a battery cell on a quantum computer. While quantum computers are still in their early, noisy stages of development, this work demonstrates a practical method for using them to solve the specific equations that govern how ions move inside a battery. The team did not simply ask a quantum machine to guess the answer; instead, they developed a hybrid approach that splits the problem between a classical computer and a quantum processor. The classical computer handles the heavy lifting of optimization, while the quantum computer performs the specific, difficult calculations required to track the movement of particles across space and time simultaneously.

The researchers focused their efforts on a specific model known as the Single Particle Model with electrolyte, which simplifies a battery down to a representative particle in the negative electrode and one in the positive electrode, surrounded by a liquid electrolyte. In a real battery, these particles are porous spheres filled with tiny channels where ions travel. The team translated the physical laws of diffusion and chemical reaction into a format that a quantum circuit could understand. They encoded the concentration of ions and the electrical potential within these particles into the quantum state of the machine, effectively creating a digital twin of the battery's internal chemistry. By using a technique that treats space and time as a single, unified block rather than calculating step-by-step, they were able to capture the entire evolution of the system in one go, avoiding the accumulation of errors that often plagues traditional step-by-step simulations.

To test their method, the team simulated a charging process where a constant electrical current was applied to the battery. They also ran a test involving a pulse of current followed by a period of rest, a common way to measure battery health. In both cases, the quantum simulation produced results that matched the expected behavior of the battery with remarkable precision. The difference between their quantum simulation and the known correct mathematical solution was incredibly small, with errors measured in the range of one part in a million to one part in a hundred thousand. This level of accuracy proves that the method works, even on the imperfect, noisy hardware available today, as demonstrated through an experiment on an emulated IBM superconducting computer. The simulation successfully tracked how the concentration of ions changed over time and space, and how the electrical potential shifted in response to the current, all while adhering to the physical laws of the system.

This achievement is notable because it represents the first time a full electrochemical cell model has been solved using a quantum algorithm. Previous attempts had focused on simpler problems, such as diffusion through a single membrane, but this work tackled the coupled, non-linear equations that describe a complete battery system. The researchers showed that by combining a specific quantum framework, known as the Feynman–Kitaev Hamiltonian, with quantum circuits designed to handle non-linear operations, they could scale the solution to include both the movement of ions and the chemical reactions at the surface of the electrode particles. They also demonstrated a new way to handle the edges of the simulation, ensuring that the ions behaved correctly at the boundaries of the particles without requiring an excessive number of complex operations.

While this is a proof of concept rather than a replacement for current battery design tools, it opens a clear path forward. The work suggests that as quantum hardware improves, this same approach could be used to model three-dimensional battery structures and more complex chemical systems that are currently impossible to simulate with classical computers. The researchers did not claim to have solved the problem of battery design forever, but they have provided a solid foundation and a working blueprint for how quantum machines can eventually tackle the most difficult problems in energy storage. By successfully running these simulations on today's hardware, they have shown that the potential of quantum computing for chemistry is not just a theoretical promise, but a practical reality that is beginning to take shape.

Technical Summary: Simulation of a Battery Cell on Quantum Computers: Reactions & Transport

Problem Statement
Simulations of electrochemical materials and systems are fundamental to technological progress but are currently constrained by the computational limits of classical methods, particularly regarding the resolution and complexity of multi-scale, three-dimensional models. While quantum computing offers the potential to overcome these limits through an exponentially large Hilbert state space, current hardware is noisy (NISQ era) and inherently performs linear, unitary operations. This creates a significant challenge in developing scalable algorithms for the non-linear partial differential equations (PDEs) that govern physical chemistry phenomena, such as diffusion and reaction rates in battery cells.

Methodology
The authors propose a hybrid quantum-classical algorithm designed to solve non-linear PDEs for electrochemical systems on current noisy hardware. The core of the approach involves:

Feynman–Kitaev (FK) Hamiltonian Formulation: The problem is cast into a spacetime formulation where the state of the system at all times and spatial points is encoded in the ground state of a single Hamiltonian. This avoids the accumulation of errors typical in time-stepping methods and allows the exponential scaling advantage to extend to the time domain.
Hybrid Optimization: To accommodate NISQ hardware, the algorithm employs a variational approach. The quantum computer evaluates a quadratic cost function based on the FK Hamiltonian, while a classical optimizer adjusts the parameters.
Quantum Nonlinear Processing Units (QNPUs): To handle non-linearities, the authors utilize QNPUs to implement operations such as multiplication by spatial coordinates ( $r$ and $1/r$ ) required for spherical coordinates.
Specific Model Implementation: The method is applied to the Single Particle Model with electrolyte (SPMe). This involves:
- Diffusion: Solving the diffusion equation for lithium-ion concentration inside spherical electrode particles using a discretized spherical Laplacian.
- Boundary Conditions: Implementing Neumann boundary conditions (zero flux at the center, fixed flux at the surface) and modifying the outer boundary to apply reactions in a small region ( $r \ge r_r$ ) rather than a single point to ensure scalability.
- Coupled Reactions: Linearizing the Butler–Volmer equation to couple the solid-phase concentration ( $c_s$ ) with the electrochemical potential ( $\phi$ ). The galvanostatic condition is enforced via an algebraic constraint coupling these variables.
Circuit Design: The quantum circuits encode the concentration in time and space using an ansatz ( $\hat{U}_c(\vec{\theta})$ ). Spatial derivatives are implemented via binary shift operators ( $\hat{A}^{(\dagger)}$ ), and time derivatives via a shift operator on an ancilla qubit to remove periodicity.

Key Contributions

First Quantum Simulation of a Full Electrochemical Cell: The paper presents the first solution for a battery cell model using a quantum algorithm, specifically the SPMe model.
Scalable Non-Linear Treatment: The work extends previous applications of the FK Hamiltonian and QNPUs (previously used for the Burgers equation) to coupled, non-linear electrochemical equations.
Improved Boundary Conditions: The authors introduce an improved, scalable treatment of boundary conditions that significantly reduces the number of required circuits and multi-qubit gates compared to existing implementations.
Coupled Variable Implementation: Unlike approaches that might solve for concentration and calculate potentials analytically in post-processing, this work deliberately implements the coupled equations for concentration and potential within the quantum algorithm to demonstrate the capability to handle complex, coupled systems.

Results
The authors successfully simulated the SPMe model under galvanostatic conditions (constant current charging) and a Galvanostatic Intermittent Titration Technique (GITT) pulse.

Accuracy: The quantum simulation results were compared against numerical solutions of the model equations. The simulations demonstrated high fidelity, with infidelity values for concentration ranging from $2.1 \times 10^{-5}$ to $4.5 \times 10^{-7}$ depending on the specific test case (charging vs. GITT).
Visualization: The results accurately captured the evolution of concentration and electrochemical potential ( $\phi = \phi_s - \phi_e$ ) in both the anode and cathode representative particles over time and space.
Hardware Context: The implementation was tested on an emulated IBM superconducting computer, validating the approach on hardware representative of current superconducting quantum processors.

Significance
The paper claims to provide a "solid foundation" for simulating more complex electrochemical systems using quantum computing. By successfully demonstrating a full battery cell model that includes both transport (diffusion) and reaction (Butler–Volmer) phenomena in a scalable, hybrid framework, the work opens the pathway for studying sophisticated three-dimensional models and differential-algebraic systems of equations that are currently at the limits of classical computational methods. The authors emphasize that while the current work is a proof of concept, the spacetime FK formulation is also compatible with future fault-tolerant quantum hardware.

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