Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum algorithms to solve partial differential equations in battery modelling

Pool, Albert, Somoza, Alejandro, Mc Keever, Conor, Lubasch, Michael, Horstmann, Birger

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

Mathematical models of electrochemical systems such as batteries or fuel cells consist of sets of coupled nonlinear partial differential equations. We describe how variational quantum algorithms (VQAs) can be used to solve these equations on a quantum computer [1]. VQAs are hybrid quantum-classical algorithms, where a cost function is calculated on a quantum computer and its parameters are optimised classically. We present a spacetime representation, inspired by the Feynman--Kitaev Hamiltonian [2], where the solution to a PDE at all times is obtained by minimising just one function. [1] A.J. Pool, A.D. Somoza, C. Mc Keever, M. Lubasch, B. Horstmann, Phys. Rev. Res. 6 (2024), 3, 033257 [2] S. Barison, F. Vicentini, J.I. Cirac, G. Carleo, Phys. Rev. Res. 4 (2022), 4, 043161

Related papers

Variational Quantum Algorithms for Continuum Modelling of Batteries

Pool, Albert, Schelling, Michael, Horstmann, Birger

Quantum computing is becoming relevant to energy science for more accurate calculations in quantum chemistry and in materials discovery with quantum machine learning methods. However, it is also applicable to cell models thanks to recent works solving nonlinear partial differential equations (PDEs). Various methods have been recently proposed, including fault-tolerant quantum linear systems algorithms (QLSAs) for linear and nonlinear PDEs. In practice, however, these fault-tolerant algorithms do not yet give a quantum advantage. On current noisy intermediate-scale quantum (NISQ) devices, these algorithms are affected by strong limitations on the depth of the quantum circuits that can be coherently executed. On NISQ devices, variational quantum algorithms (VQAs) have become a popular approach to solving PDEs. Many algorithms have been proposed that solve one time step at a time, which inherently presents a limitation to the scalability and accuracy. In our paper, we combine quantum nonlinear processing units (QNPUs) with the Feynman-Kitaev Hamiltonian to achieve a spacetime solution instead, and apply it to the nonlinear Burgers equation. We find the ground state of this Hamiltonian using an adaptive optimization procedure to avoid barren plateaus. Here, we also apply this method to an adapted version of the single particle model with electrolyte (SPMe).

•Jan 1, 2025•0 citations

Towards Solving Transport Equations in Battery Modelling with Variational Quantum Algorithms

Schelling, Michael, Pool, Albert, Steffen, David +1 more

The material design for batteries in industry and research is greatly facilitated by computational simulations on classical computers. A central tool are numerical solvers for Partial Differential Equations (PDEs), which arise as the transport equations of ions in the continuum-model of battery cells. However, these methods reach their limits when simulating the fine-grained heterogeneous structures of porous electrodes. Quantum computing is a promising candidate to achieve an advantage in scalability through the exponential growth of representable states. On current NISQ-devices promising candidates are Variational Quantum Algorithms (VQAs) to solve nonlinear PDEs. Experimental results demonstrate the capabilities of both space-time encodings with Quantum Non-linear Processing Units (QNPUs) and spectral feature mappings with Differentiable Quantum Circuits (DQCs).

•Apr 23, 2025•0 citations

Influence of different feature maps on solving partial differential equations on quantum computers

Steffen, David, Schelling, Michael, Schwab, Felix Konrad +1 more

Differentiable quantum circuits (DQCs) [1] are variational algorithms to solve partial differential equations on quantum computers. We investigate the potential of this method to solve systems of coupled partial differential equations as they occur in the simulation of electrochemical systems, e.g., fuel cells and batteries. A crucial part of DQCs is the feature space in which the input variables are encoded into quantum states. Possible choices are a Chebyshev feature map or a Fourier feature map, that generate a set of corresponding basis functions to fit the desired model. We show results on the influence of different feature maps on the expressibility and trainability for spatiotemporal models, on the use case of transport equations from battery simulation. [1] Kyriienko, O. et al., Phys. Rev. A 2021, 103, 052416

•Jan 1, 2025•0 citations

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

Albert J. Pool, Michael Schelling, Birger Horstmann

Simulations of electrochemical materials and systems accelerate technological progress, but are still limited by computational power. In particular, quantum computing offers prospects for higher resolutions, due to the exponential amount of data that can be stored in a quantum state. As current quantum computers are still noisy, we consider a hybrid quantum-classical algorithm, that divides the problem into smaller computational tasks. We describe how to implement such an algorithm for non-linear partial differential equations, the Feynman-Kitaev Hamiltonian, in a scalable way for an electrochemical system. We show how it can evaluate general electrochemical models and present a quantum simulation of the Single Particle Model with electrolyte (SPMe) as the first quantum simulation of a battery cell.

physics.chem-phCross-listedarXiv:2609.28369•Sep 24, 2026•0 citations

Quantum variational solving of nonlinear and multidimensional partial differential equations

Abhijat Sarma, Thomas W. Watts, Mudassir Moosa +2 more

A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. [Phys. Rev. A 101, 010301(R) (2020)]. In this paper we generalize the method introduced by Lubasch et al. to cover a broader class of nonlinear PDEs as well as multidimensional PDEs and study the performance of the variational quantum algorithm on several example equations. Specifically, we show via numerical simulations that the algorithm can solve instances of the single-asset Black-Scholes equation with a nontrivial nonlinear volatility model, the double-asset Black-Scholes equation, the Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our simulations use up to $n=12$ Ansatz qubits, computing PDE solutions with ${2}^{n}$ grid points. We also perform proof-of-concept demonstrations with a trapped-ion quantum processor from IonQ [Nat. Commun. 10, 5464 (2019)], showing accurate computation of two representative expectation values needed for the calculation of a single time step of the nonlinear Black-Scholes equation. Through our classical simulations and demonstrations on quantum hardware, we identify and discuss several open challenges for using quantum variational methods to solve PDEs in a regime with a large number (much greater than ${2}^{20}$) of grid points, but also a practical number of gates per circuit and circuit shots.

arXiv:2311.01531•Physical Review A•Jun 21, 2024•38 citations

Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer

Mazen Ali, Matthias Kabel

Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate-scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE---the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices---especially when considering large system sizes and more complicated nonlinear PDEs that are required in order to be competitive with classical high-end solvers.

arXiv:2211.14064•Physical Review Applied•Jul 25, 2023•30 citations
Mathematical models of electrochemical systems such as batteries or fuel cells consist of sets of coupled nonlinear partial differential equations. We describe how variational quantum algorithms (VQAs) can be used to solve these equations on a quantum computer [1]. VQAs are hybrid quantum-classical algorithms, where a cost function is calculated on a quantum computer and its parameters are optimised classically. We present a spacetime representation, inspired by the Feynman--Kitaev Hamiltonian [2], where the solution to a PDE at all times is obtained by minimising just one function. [1] A.J. Pool, A.D. Somoza, C. Mc Keever, M. Lubasch, B. Horstmann, Phys. Rev. Res. 6 (2024), 3, 033257 [2] S. Barison, F. Vicentini, J.I. Cirac, G. Carleo, Phys. Rev. Res. 4 (2022), 4, 043161
Quantum computing is becoming relevant to energy science for more accurate calculations in quantum chemistry and in materials discovery with quantum machine learning methods. However, it is also applicable to cell models thanks to recent works solving nonlinear partial differential equations (PDEs). Various methods have been recently proposed, including fault-tolerant quantum linear systems algorithms (QLSAs) for linear and nonlinear PDEs. In practice, however, these fault-tolerant algorithms do not yet give a quantum advantage. On current noisy intermediate-scale quantum (NISQ) devices, these algorithms are affected by strong limitations on the depth of the quantum circuits that can be coherently executed. On NISQ devices, variational quantum algorithms (VQAs) have become a popular approach to solving PDEs. Many algorithms have been proposed that solve one time step at a time, which inherently presents a limitation to the scalability and accuracy. In our paper, we combine quantum nonlinear processing units (QNPUs) with the Feynman-Kitaev Hamiltonian to achieve a spacetime solution instead, and apply it to the nonlinear Burgers equation. We find the ground state of this Hamiltonian using an adaptive optimization procedure to avoid barren plateaus. Here, we also apply this method to an adapted version of the single particle model with electrolyte (SPMe).
The material design for batteries in industry and research is greatly facilitated by computational simulations on classical computers. A central tool are numerical solvers for Partial Differential Equations (PDEs), which arise as the transport equations of ions in the continuum-model of battery cells. However, these methods reach their limits when simulating the fine-grained heterogeneous structures of porous electrodes. Quantum computing is a promising candidate to achieve an advantage in scalability through the exponential growth of representable states. On current NISQ-devices promising candidates are Variational Quantum Algorithms (VQAs) to solve nonlinear PDEs. Experimental results demonstrate the capabilities of both space-time encodings with Quantum Non-linear Processing Units (QNPUs) and spectral feature mappings with Differentiable Quantum Circuits (DQCs).
Differentiable quantum circuits (DQCs) [1] are variational algorithms to solve partial differential equations on quantum computers. We investigate the potential of this method to solve systems of coupled partial differential equations as they occur in the simulation of electrochemical systems, e.g., fuel cells and batteries. A crucial part of DQCs is the feature space in which the input variables are encoded into quantum states. Possible choices are a Chebyshev feature map or a Fourier feature map, that generate a set of corresponding basis functions to fit the desired model. We show results on the influence of different feature maps on the expressibility and trainability for spatiotemporal models, on the use case of transport equations from battery simulation. [1] Kyriienko, O. et al., Phys. Rev. A 2021, 103, 052416
Simulations of electrochemical materials and systems accelerate technological progress, but are still limited by computational power. In particular, quantum computing offers prospects for higher resolutions, due to the exponential amount of data that can be stored in a quantum state. As current quantum computers are still noisy, we consider a hybrid quantum-classical algorithm, that divides the problem into smaller computational tasks. We describe how to implement such an algorithm for non-linear partial differential equations, the Feynman-Kitaev Hamiltonian, in a scalable way for an electrochemical system. We show how it can evaluate general electrochemical models and present a quantum simulation of the Single Particle Model with electrolyte (SPMe) as the first quantum simulation of a battery cell.
A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. [Phys. Rev. A 101, 010301(R) (2020)]. In this paper we generalize the method introduced by Lubasch et al. to cover a broader class of nonlinear PDEs as well as multidimensional PDEs and study the performance of the variational quantum algorithm on several example equations. Specifically, we show via numerical simulations that the algorithm can solve instances of the single-asset Black-Scholes equation with a nontrivial nonlinear volatility model, the double-asset Black-Scholes equation, the Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our simulations use up to n=12n=12n=12 Ansatz qubits, computing PDE solutions with 2n{2}^{n}2n grid points. We also perform proof-of-concept demonstrations with a trapped-ion quantum processor from IonQ [Nat. Commun. 10, 5464 (2019)], showing accurate computation of two representative expectation values needed for the calculation of a single time step of the nonlinear Black-Scholes equation. Through our classical simulations and demonstrations on quantum hardware, we identify and discuss several open challenges for using quantum variational methods to solve PDEs in a regime with a large number (much greater than 220{2}^{20}220) of grid points, but also a practical number of gates per circuit and circuit shots.
Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate-scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE---the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices---especially when considering large system sizes and more complicated nonlinear PDEs that are required in order to be competitive with classical high-end solvers.