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

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

Steffen, David, Schelling, Michael, Schwab, Felix Konrad, Horstmann, Birger

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

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

Related papers

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

Solving nonlinear differential equations with differentiable quantum circuits

Oleksandr Kyriienko, Annie E. Paine, Vincent E. Elfving

We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.

arXiv:2011.10395•Physical Review A•May 17, 2021•185 citations

Quantum algorithms to solve partial differential equations in battery modelling

Pool, Albert, Somoza, Alejandro, Mc Keever, Conor +2 more

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

•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

Protocols for trainable and differentiable quantum generative modeling

Oleksandr Kyriienko, Annie E. Paine, Vincent E. Elfving

We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modeling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with the proposed phase feature map of exponential capacity. This is followed by a trainable quantum circuit, forming the model. We then map the trained model to the bit basis using a fixed unitary transformation, in this case corresponding to a quantum Fourier transform circuit. It allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Our approach opens a route to multidimensional generative modeling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubitwise training and feature map sparsification. Published by the American Physical Society 2024

arXiv:2202.08253•Physical Review Research•Sep 12, 2024•14 citations

Quantum Model-Discovery

Niklas Heim, Atiyo Ghosh, Oleksandr Kyriienko +1 more

Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) efficiently. Recently, variational approaches to solving non-linear PDEs also with near-term quantum devices were proposed. One of the most promising general approaches is based on recent developments in the field of scientific machine learning for solving PDEs. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks, including the discovery of differential equations from a dataset of measurements. We use differentiable quantum circuits (DQCs) to solve equations parameterized by a library of operators, and perform regression on a combination of data and equations. Our results show a promising path to Quantum Model Discovery (QMoD), on the interface between classical and quantum machine learning approaches. We demonstrate successful parameter inference and equation discovery using QMoD on different systems including a second-order, ordinary differential equation and a non-linear, partial differential equation.

arXiv:2111.06376•Nov 11, 2021•3 citations
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
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).
We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.
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
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.
We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modeling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with the proposed phase feature map of exponential capacity. This is followed by a trainable quantum circuit, forming the model. We then map the trained model to the bit basis using a fixed unitary transformation, in this case corresponding to a quantum Fourier transform circuit. It allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Our approach opens a route to multidimensional generative modeling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubitwise training and feature map sparsification. Published by the American Physical Society 2024
Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) efficiently. Recently, variational approaches to solving non-linear PDEs also with near-term quantum devices were proposed. One of the most promising general approaches is based on recent developments in the field of scientific machine learning for solving PDEs. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks, including the discovery of differential equations from a dataset of measurements. We use differentiable quantum circuits (DQCs) to solve equations parameterized by a library of operators, and perform regression on a combination of data and equations. Our results show a promising path to Quantum Model Discovery (QMoD), on the interface between classical and quantum machine learning approaches. We demonstrate successful parameter inference and equation discovery using QMoD on different systems including a second-order, ordinary differential equation and a non-linear, partial differential equation.