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 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.
Abstract This paper introduces a quantum-enhanced finite element method (FEM) designed for noisy intermediate-scale quantum (NISQ) devices, leveraging variational quantum algorithms (VQAs) to solve engineering partial differential equations. We demonstrate the framework by solving the Euler–Bernoulli beam and the NAFEMS T4 heat transfer problems, which involve Dirichlet, Neumann, and Robin boundary conditions. A key innovation is a ‘set-to-zero’ strategy that incorporates boundary conditions through a correction matrix, K bc , allowing for flexible imposition at any node without domain decomposition. The global stiffness matrix is decomposed into a constant number of Pauli terms, O (1), using the method by Sato et al while boundary terms are handled with a sublinearly scaling partial Pauli measurement technique. The algorithm achieves logarithmic qubit scaling ( n = ⌈ log 2 N ⌉ qubits for N degrees of freedom(DOF)) and employs shallow, hardware-efficient circuits with empirically trainable depth for small-scale systems. Validation on the Qiskit statevector simulator shows high accuracy. For the Euler–Bernoulli beam problem with 4 to 64 DOF, the algorithm achieves relative errors of 0.5%–1.5% and fidelities of 0.998–0.999. For the NAFEMS T4 heat transfer benchmark, a 5.4% relative error is observed. The VQA converges robustly within 77–350 iterations, though barren plateaus are a known challenge for scaling to larger systems. This work establishes a scalable framework for quantum FEM, offering a significant memory advantage over classical methods and advancing the potential for quantum-enhanced engineering simulations.
This article proposes a variational quantum algorithm to solve linear and nonlinear thermofluid dynamic transport equations. The hybrid classical-quantum framework is applied to problems governed by the heat, wave, and Burgers’ equations in combination with different engineering boundary conditions. Topics covered include the encoding of band matrices, as in the consideration of nonconstant material properties and upwind-biased first- and higher-order approximations, widely used in engineering computational fluid dynamics, by the use of a mask function. Verification examples demonstrate high predictive agreement with classical methods. Furthermore, the scalability analysis shows a polylog scaling of the number of quantum gates with the number of qubits. Remaining challenges refer to the implicit construction of upwind schemes and the identification of an appropriate parameterization strategy of the quantum ansatz.
Abstract We propose an explicit quantum framework for numerically simulating general linear partial differential equations (PDEs), extending previous work (Guseynov et al 2025 Phys. Rev. Res. 7 033100) to incorporate (a) Robin boundary conditions—which include Neumann and Dirichlet conditions as special cases–(b) inhomogeneous terms, and (c) variable coefficients in space and time. Our approach begins with a general finite-difference discretization and applies the Schrödingerisation technique to transform the resulting system into one that admits unitary quantum evolution, enabling quantum simulation. For the Schrödinger equation corresponding to the discretized PDE, we construct an efficient block-encoding of the Hamiltonian H that scales polylogarithmically with the number of grid points N . This encoding is compatible with quantum signal processing and allows for the implementation of the evolution operator e − i H t . The explicit circuit construction in our method permits complexity to be measured in fundamental gate units–namely, CNOT gates and single-qubit rotations–bypassing the inefficiencies of oracle queries. Consequently, the overall algorithm scales polynomially with N and linearly with the spatial dimension d . Under certain input/output assumptions our method achieves a polynomial speedup in N and an exponential advantage in d for a wide class of PDEs, thereby mitigating the classical curse of dimensionality. The validity and efficiency of the proposed approach are further substantiated by numerical simulations. By explicitly defining the quantum operations and quantifying their resource requirements, our approach offers a practical alternative for numerically solving PDEs, distinct from others that rely on oracle queries and purely asymptotic scaling methods.
This paper proposes a variational quantum algorithm to address the large-scale and low computational efficiency issues of high-dimensional partial differential equations (PDEs) in climate science. In this method, parameterized quantum circuits encode the discrete solutions of PDEs into quantum states. Then, through repeated quantum measurements, we can obtain the expectation values of the operators. By iteratively adjusting the circuit parameters, a hybrid quantum-classical optimization loop can reduce the error in the solution. The Schrödinger equation and the Poisson equation are benchmark partial differential equations for numerical experiments, and they are four-dimensional and five-dimensional, respectively. According to the above results, the convergence speed is relatively fast; after 100 iterations, the median relative errors of all test cases are less than 1.6% and 3.1%, respectively, with the final target loss ranging between 0.014 and 0.045. The above comparison indicates that in high-dimensional and ill-posed problems, quantum solvers can achieve the same or better accuracy and robustness compared to traditional methods. Resource analysis indicates that quantum algorithms use less memory and fewer sub-exponential growth gates and runtime in larger systems. According to the above findings, variational quantum solvers can be used for complex scientific computing problems. In addition, they hope that future climate models will be more accurate and efficient.
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.
Global variational quantum algorithm (VQA) frameworks for solving partial differential equations (PDEs) often rely on a single expressive ansatz over a uniform grid, which becomes structurally inefficient when solutions exhibit spatially heterogeneous complexity such as localized singularities or thin boundary layers. A localized nonsmooth feature can degrade the convergence of the entire global quantum representation, imposing unnecessary circuit depth and amplifying the risk of vanishing gradients in barren plateaus.
To overcome this limitation, we propose a block-wise VQA framework for PDEs characterized by spatially heterogeneous solution complexity. The computational domain is decomposed into locally represented quantum subproblems, where grid budgets and ansatz families are dynamically assigned based on a computable difficulty indicator. Artificial block interfaces are coupled through unified jump penalties for state values, derivatives, and physical fluxes, while adaptive reblocking tracks moving rough regions to maintain local accuracy without excessive global qubit overhead. The error analysis rigorously separates spatial discretization, ansatz expressivity, optimization convergence, finite-shot sampling noise, transfer errors from reblocking, and interface-coupling contributions. Reproducible residual-based simulations on representative elliptic, advection--diffusion, and Burgers problems demonstrate lower approximation errors and reduced peak local circuit width compared to global VQA approaches. These results substantiate block-wise quantum resource localization, confirming that high-fidelity PDE solutions can be achieved on near-term quantum devices without requiring a single globally expressive ansatz.