Boundary value problems (BVPs) arise in various scientific and engineering disciplines, including physics, finance, and biology. Classical computers may not be able to solve these problems quickly or accurately due to their computational complexity. Quantum Fourier transform (QFT) may efficiently solve differential equations including the Schrödinger equation and Helmholtz equation, according to recent quantum computing research. Quantum annealing determines a Hamiltonian's ground state using quantum fluctuation. The quantum computing BVP approach is novel to numerical methods. Quantum methods like quantum phase estimation have not been employed to solve BVPs in heat transfer, groundwater modelling, or advection diffusion response. The Harrow-Hassidim-Lloyd (HHL) quantum algorithm solves steady-state heat and radon-diffusion equations in this research study. The BVP linear operator eigenvalues are computed using the quantum phase estimation approach. We used IBM's 'Qiskit' package to compute in Python. IBM Qiskit's 'Linear Solver' program provided the quantum computing numerical solution. Tridiagonal Toeplitz matrix role is shown. Comparisons with traditional results and future applications were made.
Boundary value problems (BVPs) arise in various scientific and engineering disciplines, including physics, finance, and biology. Classical computers may not be able to solve these problems quickly or accurately due to their computational complexity. Quantum Fourier transform (QFT) may efficiently solve differential equations including the Schrödinger equation and Helmholtz equation, according to recent quantum computing research. Quantum annealing determines a Hamiltonian's ground state using quantum fluctuation. The quantum computing BVP approach is novel to numerical methods. Quantum methods like quantum phase estimation have not been employed to solve BVPs in heat transfer, groundwater modelling, or advection diffusion response. The Harrow-Hassidim-Lloyd (HHL) quantum algorithm solves steady-state heat and radon-diffusion equations in this research study. The BVP linear operator eigenvalues are computed using the quantum phase estimation approach. We used IBM's 'Qiskit' package to compute in Python. IBM Qiskit's 'Linear Solver' program provided the quantum computing numerical solution. Tridiagonal Toeplitz matrix role is shown. Comparisons with traditional results and future applications were made.
Quantum computing is coming up as futuristic technology in engineering and science.The famous Sturm-Liouville (SL) differential equation along with some specific boundary conditions appear in many branches of physics as well as in other fundamental fields.The numerical solution of such boundary value problems (like heat transfer, potential problem etc.) is frequently computationally expensive, and traditional computers may not be able to deliver accurate solutions in an acceptable length of time.We propose quantum computing approach to solve such boundary value problems that will be more efficient and time solvent.The present research demands that the Harrow-Hassidim-Lloyd (HHL) quantum algorithm provides an efficient technique for solving such boundary value problems that arise in many branches of physical science and engineering.The prime focus of this research is to explore HHL quantum algorithm in detail.The quantum phase estimation algorithm is applied for solving eigen values and eigenvectors associated with the SL type of boundary value problem.As a case study, in this research, we have solved the steady state heat transfer problem with boundary conditions.Our computation is based on Qiskit module made by IBM and available as a library in Python.A detailed analysis of the computational complexity of our algorithm that compares its performance with classical methods on a range of boundary value problems has been discussed.Role of Tridiagonal Toeplitz matrix is revealed.Results demonstrate that the method of quantum algorithm for solving boundary value problems can significantly outperform classical methods, especially for larger problem sizes.This work represents a significant methodology in solving important equations of physics and mathematics in future quantum computers.
Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.
Researchers are currently working on computational solutions based on quantum systems to accelerate the speed of complex mathematical models. This work presented how to formulate complex computational problems as a quantum system of linear equations and find solutions using Quantum Linear System Algorithm (QLSA), also called Quantum Harrow-Hassidim-Lloyd (HHL) algorithm. This paper showed experimental evaluation of multiple problem statements (curve-fitting functions, interpolating polynomials) as a quantum system of linear equations that involve computation of Vandermonde matrices as co-efficient matrices on IBM Quantum Information Software Kit for Quantum Computation (QISKit) platform. Along with a few examples demonstrating its evaluation on diagonal, Hermitian, and Non-Hermitian matrices as co-efficient matrices. The fidelity is used as a measure of performance for comparing the accuracy of quantum results with respect to existing classical solutions on IBM QISKit and drawing conclusions from the experimental results. Experimental evaluation shows that the fidelity depends on the sparsity of the input matrices and therefore the results vary depending on those matrices.
Time-fractional diffusion equations have emerged as powerful models for describing anomalous transport phenomena in physics, biology and engineering. To address the computational challenges arising from their non-local operators, we employ the WEB-spline finite element method, which provides a flexible and accurate discretization framework. The resulting linear system of equations are then explored in the context of quantum computing. Specifically, we investigate three prominent quantum linear solvers: the variational quantum linear solver (VQLS), the Harrow-Hassidim-Lloyd (HHL) algorithm and quantum annealing (QA). VQLS leverages hybrid variational techniques and shallow circuits, making it well-suited for noisy intermediate-scale quantum (NISQ) devices, while HHL offers a theoretically exponential speedup for sparse systems but requires deep fault-tolerant circuits. QA, in contrast, reformulates the problem into a quadratic unconstrained binary optimization (QUBO) instance, enabling approximate solutions through energy minimization on specialized hardware. We present a comparative analysis in terms of circuit depth, noise resilience, scalability and solution extraction, including the role of quantum state tomography in reconstructing classical information. Numerical experiments on a time-fractional diffusion problem highlight the complementary strengths and limitations of each method. This study bridges advanced numerical discretization with emerging quantum algorithms, providing insights into the feasibility and future potential of quantum-enhanced solvers for fractional partial differential equations.
Quantum computing promises a theoretical computational advantage for solving certain problems, thus it holds significant academic and commercial interest. This thesis begins by reviewing established quantum algorithms and their applications. Then I expand this set to demonstrate quantum computational advantage for solving the drift-diffusion equation, a computationally intensive problem with applications in finance, modelling, and manufacturing. First, I introduce a complexity analysis for four classical algorithms designed to solve the multi-dimensional drift-diffusion equation. I then present four quantum algorithms, leveraging a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. By comparing the complexities of these methods to their classical counterparts, I find that diagonalisation via the quantum Fourier transform offers a quantum computational advantage for solving linear partial differential equations at a fixed final time. I employ a multidimensional amplitude estimation process to extract the full probability distribution from the quantum computer. Finally, I compare the circuit depths for five gate sets to implement a quantum algorithm solving the drift-diffusion equation in two spatial dimensions. This uses the diagonalisation by the quantum Fourier transform algorithm. The gate sets are: an unconstrained gate set, TK1 gate set from Quantinuum, the native gate sets of IBM Heron and IonQ, and Fujitsu’s space-time efficient analog rotation (STAR) gate set. This analysis covers a set of illustrative scenarios using up to 22 qubits. I find that while scaling with spatial resolution aligns with theoretical predictions in one dimension, scaling with spatial dimension is less efficient than theorised due to overhead from block encoding. Finally, using the STAR gate set, I find that even minimal problem instances exceed the operational limits of current quantum hardware. We find that for a test scenario using 22 logical qubits, requires 1 × 106 logical operations, assuming full connectivity.
Quantum methods for solving partial differential equations (PDEs) generally fall into a few general categories, each leveraging different quantum mechanisms to achieve potential speedups. One well-known approach is the quantum linear system solver, epitomized by the Harrow-Hassidim-Lloyd (HHL) algorithm, which uses phase estimation and eigenvalue inversion to solve large, sparse linear systems arising from discretized linear PDEs [1]. This method can yield exponential improvements over classical solvers under certain conditions (mostly when the matrix is well-conditioned and sparse). More recent developments, such as Quantum Singular Value Transformation (QSVT), provide a framework for matrix inversion in a more flexible way, often improving on the cost and error dependence of earlier solvers [2]. Time-dependent PDEs can be tackled via Hamiltonian simulations. The Trotter-Suzuki method decomposes the exponential of a Hamiltonian into products of exponentials of simpler sub-Hamiltonians, allowing a piecewise approximation of continuous dynamics [3]. Continuous-time quantum walks, which treat a PDE operator as a Hamiltonian that governs the walk, similarly simulate processes like diffusion or wave propagation on discretized grids or graphs [4]. Beyond direct simulation, variational quantum algorithms have emerged as a promising route in the noisy intermediate-scale quantum era. These approaches encode the PDE solution as a parameterized quantum state and optimize the parameters via classical feedback, thereby minimizing a cost function tied to the PDE residual or energy functional [5]. Quantum annealing or adiabatic methods, meanwhile, can solve optimization problems derived from PDE constraints by adiabatically evolving a simple initial Hamiltonian to the final problem Hamiltonian and staying in the system's ground state [3]. In our study, we implement a new parameterized quantum model, specifically designed to solve large linear systems on quantum hardware. This model is less deep and computationally faster than the proposed one in [5], reducing the complexity and speeding up convergence times. Such improvements are particularly valuable in cases where classical algorithms may fail due to the problem's size or complexity. Ongoing research is focusing on generalized quantum models that can be adapted to practical PDE problems for solving large electromagnetic problems where classical computers can fail.