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).
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).
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).
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.
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
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
Numerical solvers for Partial Differential Equations (PDEs) are of great interest in various domains, e.g., in aerodynamics or for transport equations in lectrochemistry [1], and the need for fine-grained solutions of on-linear PDEs is growing. While subspace methods allow for a dimensionality reduction, non-linear PDEs require linearization schemes, such as the Carleman-linearization [2], resulting in linear systems of exponential dimensionality, operating on the limits of classical methods. Motivated by Krylov-subspace methods [3], which find approximate solutions in iteratively growing subspaces, the aim of this talk is to investigate the potential of two existing methods from quantum computing to compose a NISQ-era hybrid quantum-classical algorithm. Firstly, non-linear quantum computing (QNPUs) [4,5] and secondly Quantum Subspace Expansion (QSE) [6], both promising tools towards more scalable computations. The use of QNPUs enables linearization in a tensor-product-subspace using ancilla qubits, while offering efficient gate-based implementations. On the other hand, QSE measures high-dimensional overlaps on a quantum computer. The combination of these two methods yields the possibility of encapsulating the high-dimensional steps of linearization and subspace projection on a quantum device. As a result, only a lower-dimensional subspace problem remains to be solved on classical hardware.
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 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) of grid points, but also a practical number of gates per circuit and circuit shots.