We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in terms of time complexity but also has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation costs scale economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering applications, such as the subdiffusion equation, the nonlinear Burgers' equation, and a coupled diffusive epidemic model. We assess quantum hardware performance under realistic noise conditions, further validating the practical utility of our algorithm.
We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in terms of time complexity but also has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation costs scale economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering applications, such as the subdiffusion equation, the nonlinear Burgers' equation, and a coupled diffusive epidemic model. We assess quantum hardware performance under realistic noise conditions, further validating the practical utility of our algorithm.
Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the Ansatz volume for gradient estimation and how the time-to-solution scales with the diffusion parameter. Our proposed algorithm extends economically to higher-order time-stepping schemes, such as the Crank-Nicolson method. We present a semi-implicit scheme for solving systems of evolution equations with non-linear terms, such as the reaction-diffusion and the incompressible Navier-Stokes equations, and demonstrate its validity by proof-of-concept results.
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.
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.
The paper presents a variational quantum algorithm to solve initial–boundary value problems described by second-order partial differential equations. The approach uses hybrid classical/quantum framework that is well suited for quantum computers of the current noisy intermediate-scale quantum era. The partial differential equation is initially translated into an optimal control problem with a modular control-to-state operator (ansatz). The objective function and its derivatives required by the optimizer can efficiently be evaluated on a quantum computer by measuring an ancilla qubit, while the optimization procedure employs classical hardware. The focal aspect of the study is the treatment of boundary conditions, which is tailored to the properties of the quantum hardware using a correction technique. For this purpose, the boundary conditions and the discretized terms of the partial differential equation are decomposed into a sequence of unitary operations and subsequently compiled into quantum gates. The accuracy and gate complexity of the approach are assessed for second-order partial differential equations by classically emulating the quantum hardware. The examples include steady and unsteady diffusive transport equations for a scalar property in combination with various Dirichlet, Neumann, or Robin conditions. The results of this flexible approach display a robust behavior and a strong predictive accuracy in combination with a remarkable polylog complexity scaling in the number of qubits of the involved quantum circuits. Remaining challenges refer to adaptive ansatz strategies that speed up the optimization procedure.
The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm leads a promising tool for solving partial differential equations on noisy intermediate scale quantum devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still at the early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of the VQLS implementation are discussed by various test instances of linear systems. The effect of the number of shots on the accuracy is studied, which reveals a logarithmic relationship. Furthermore, the heuristic scaling of the VQLS with the precision ε, the number of qubits n and the condition number k validates its time complexity reported in the literature. In addition, the successful state vector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present VQLS-based algorithm by proof-of-concept results. Finally, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision ε and linearly dependent on the number of qubits n.
Solving linear systems of equations is central to many engineering and
scientific fields. Several quantum algorithms have been proposed for linear
systems, where the goal is to prepare ∣x⟩ such that A∣x⟩∝∣b⟩. While these algorithms are promising, the time horizon for their
implementation is long due to the required quantum circuit depth. In this work,
we propose a variational hybrid quantum-classical algorithm for solving linear
systems, with the aim of reducing the circuit depth and doing much of the
computation classically. We propose a cost function based on the overlap
between ∣b⟩ and A∣x⟩, and we derive an operational meaning for
this cost in terms of the solution precision ϵ. We also introduce a
quantum circuit to estimate this cost, while showing that this cost cannot be
efficiently estimated classically. Using Rigetti's quantum computer, we
successfully implement our algorithm up to a problem size of 32×32.
Furthermore, we numerically find that the complexity of our algorithm scales
efficiently in both 1/ϵ and κ, with κ the condition
number of A. Our algorithm provides a heuristic for quantum linear systems
that could make this application more near term.