For the solution of partial differential equations (PDEs), we show that the quantum Fourier transform (QFT) can enable the design of quantum circuits that are particularly simple, both conceptually and with regard to hardware requirements. This is shown by explicit circuit constructions for the incompressible advection, heat, isotropic acoustic wave, and Poisson's equations as canonical examples. We utilize quantum singular value transformation to develop circuits that are expected to be of optimal computational complexity. Additionally, we consider approximations suited for smooth initial conditions and describe circuits that make lower demands on hardware. The simple QFT-based circuits are efficient with respect to dimensionality and pave the way for current quantum computers to solve high-dimensional PDEs.
For the solution of partial differential equations (PDEs), we show that the quantum Fourier transform (QFT) can enable the design of quantum circuits that are particularly simple, both conceptually and with regard to hardware requirements. This is shown by explicit circuit constructions for the incompressible advection, heat, isotropic acoustic wave, and Poisson's equations as canonical examples. We utilize quantum singular value transformation to develop circuits that are expected to be of optimal computational complexity. Additionally, we consider approximations suited for smooth initial conditions and describe circuits that make lower demands on hardware. The simple QFT-based circuits are efficient with respect to dimensionality and pave the way for current quantum computers to solve high-dimensional PDEs.
Quantum algorithms offer new avenues for solving partial differential equations (PDEs). While the potential for end-to-end quantum advantage is at present not well understood, recent literature presents explicit circuit constructions for solving certain classes of linear PDEs in the frequency domain and thus offers concrete examples to study. In this work, we develop end-to-end implementations of these quantum circuits compiled to machine-level instructions and benchmark them in both numerical simulations and IBMQ hardware experiments. We focus on the advection, wave, and Poisson equations and study quantum circuits that propagate the dynamics in frequency space via the quantum Fourier transform using approximate methods based on a first-order approximation which offer compact representations with uncontrollable approximation error, and polynomial approximation methods based on quantum signal processing (QSP) leading to deeper circuits with tunable algorithmic error. In addition, we experimentally demonstrate that the QSP-augmented algorithm can provide accurate solutions under realistic hardware constraints. Finally, we extend our method to address non-homogeneous Dirichlet boundary conditions and verify it numerically for a Poisson equation with source term obtained from high-fidelity physics simulations of a capacitively coupled plasma.
Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schrödinger-type equations. To address this limitation, Schrödingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schrödinger-type equations. However, despite the development of Schrödingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schrödingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.
Quantum computing has the potential to revolutionize computational problem-solving by leveraging the quantum mechanical phenomena such as superposition and entanglement, enabling fundamentally different computational strategies compared to classical computing. Classical methods for solving complex and high dimensional partial differential equations (PDEs), crucial for modeling diverse physical phenomena, suffer from an exponential growth in computational complexity known as the curse of dimensionality. This can limit their effectiveness for certain high-resolution simulations and real-time computational applications. In contrast, quantum computing inherently exploits superposition and entanglement to efficiently represent and simultaneously manipulate large-scale discretized PDEs, using exponentially fewer resources compared to classical systems. There are currently many quantum techniques available for solving partial differential equations (PDEs), which are mainly based on variational quantum circuits. However, the existing quantum PDE solvers, particularly those based on variational quantum eigensolver (VQE) techniques, suffer from several limitations. These include low accuracy, high execution times, and low scalability on quantum simulators as well as on noisy intermediate-scale quantum (NISQ) devices, especially for multidimensional PDEs. In this work, we propose an efficient and scalable algorithm for solving multidimensional PDEs. We present two variants of our algorithm: the first leverages finite-difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, while the second employs FDM, C2Q, and column-by-column decomposition (CCD). Both variants are designed to enhance accuracy and scalability while reducing execution times. We have validated and evaluated our proposed concepts using a number of case studies including multidimensional Poisson equation, multidimensional heat equation, Black Scholes equation for financial option pricing, and Navier-Stokes equation for computational fluid dynamics (CFD), achieving promising results. Our results demonstrate higher accuracy, higher scalability, and faster execution times compared to VQE-based solvers on noise-free and noisy quantum simulators from IBM. Additionally, we validated our approach on hardware emulators and actual quantum hardware, employing noise mitigation techniques. This work establishes a practical and effective approach for solving PDEs using quantum computing for engineering and scientific applications.
Contemporary quantum computers open up novel possibilities for tackling intricate problems, encompassing quantum system modeling and solving partial differential equations (PDEs). This paper explores the optimization of quantum algorithms aimed at resolving PDEs, presenting a significant challenge within the realm of computational science. The work delves into the application of the Variational Quantum Eigensolver (VQE) for addressing equations such as Poisson's equation. It employs a Hamiltonian constructed using a modified Feynman-Kitaev formalism for a VQE, which represents a quantum system and encapsulates information pertaining to the classical system. By optimizing the parameters of the quantum circuit that implements this Hamiltonian, it becomes feasible to achieve minimization, which corresponds to the solution of the original classical system. The modification optimizes quantum circuits by minimizing the cost function associated with the VQE. The efficacy of this approach is demonstrated through the illustrative example of solving the Poisson equation. The prospects for its application to the integration of more generalized PDEs are discussed in detail. This study provides an in-depth analysis of the potential advantages of quantum algorithms in the domain of numerical solutions for the Poisson equation and emphasizes the significance of continued research in this direction. By leveraging quantum computing capabilities, the development of more efficient methodologies for solving these equations is possible, which could significantly transform current computational practices. The findings of this work underscore not only the practical advantages but also the transformative potential of quantum computing in addressing complex PDEs. Moreover, the results obtained highlight the critical need for ongoing research to refine these techniques and extend their applicability to a broader class of PDEs, ultimately paving the way for advancements in various scientific and engineering domains.
In this work we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schr\"odinger-type, Hamiltonian partial differential equations (PDEs), using space-efficient variational circuits, including the symmetries of the problem, and global and gradient-based optimizers. We use this algorithm to benchmark the performance of the representation techniques by means of the computation of the ground state in three PDEs, i.e., the one-dimensional quantum harmonic oscillator and the transmon and flux qubits, studying how they would perform in ideal and near-term quantum computers. With the Fourier methods developed here, we obtain low infidelities of order ${10}^{\ensuremath{-}4}--{10}^{\ensuremath{-}5}$ using only three to four qubits, demonstrating the high compression of information in a quantum computer. Practical fidelities are limited by the noise and the errors of the evaluation of the cost function in real computers, but they can also be improved through error mitigation techniques.
Abstract Many real-world problems, like modelling environment dynamics, physical processes, time series etc involve solving partial differential equations (PDEs) parameterised by problem-specific conditions. Recently, a deep learning architecture called Fourier neural operator (FNO) proved to be capable of learning solutions of given PDE families for any initial conditions as input. However, it results in a time complexity linear in the number of evaluations of the PDEs while testing. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we exploit the running efficiency offered by these and propose quantum algorithms inspired by the classical FNO, which result in time complexity logarithmic in the number of evaluations and are expected to be substantially faster than their classical counterpart. At their core, we use the unary encoding paradigm and orthogonal quantum layers and introduce a new quantum Fourier transform in the unary basis. We propose three different quantum circuits to perform a quantum FNO. The proposals differ in their depth and their similarity to the classical FNO. We also benchmark our proposed algorithms on three PDE families, namely Burgers’ equation, Darcy’s flow equation and the Navier–Stokes equation. The results show that our quantum methods are comparable in performance to the classical FNO. We also perform an analysis on small-scale image classification tasks where our proposed algorithms are at par with the performance of classical convolutional neural networks, proving their applicability to other domains as well.