We present four quantum algorithms for solving a multidimensional drift-diffusion equation. They rely on a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. We compare the complexities of these methods to their classical counterparts, finding that diagonalization via the quantum Fourier transform offers a quantum computational advantage for solving linear partial differential equations at a fixed final time. We employ a multidimensional amplitude estimation process to extract the full probability distribution from the quantum computer.
We present four quantum algorithms for solving a multidimensional drift-diffusion equation. They rely on a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. We compare the complexities of these methods to their classical counterparts, finding that diagonalization via the quantum Fourier transform offers a quantum computational advantage for solving linear partial differential equations at a fixed final time. We employ a multidimensional amplitude estimation process to extract the full probability distribution from the quantum computer.
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.
Partial differential equations (PDEs) play a central role in scientific and engineering analysis, with applications spanning fluid dynamics, heat and mass transfer, electromagnetism, quantum mechanics, and financial modeling, where they are used to describe diffusion processes, wave propagation, and the evolution of complex systems over space and time. Solving multidimensional partial differential equations (PDEs) is a computationally challenging problem, even for the most advanced classical systems. Over the past decade, quantum computing has attracted significant interest as a potential approach for solving complex computational problems, including multidimensional PDEs. Although a variety of approaches have been proposed for solving PDEs, most of the existing techniques are based on variational quantum algorithms (VQAs). Despite being promising, these VQA-based approaches suffer from low accuracy, long execution times, and limited scalability. In this work, we propose a scalable and efficient quantum algorithm for solving multidimensional PDEs. Our algorithm has two variants. One variant is based on the finite difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, whereas the other is based on FDM, C2Q, and column-by-column decomposition (CCD). We have also evaluated our algorithm using several practical case studies; namely, Poisson, heat, Black–Scholes, and Navier–Stokes equations. The results show that our proposed approach achieves higher accuracy, greater scalability, and faster execution time than the VQA-based approaches. We validated these findings on both noise-free and noisy simulators, as well as on a hardware emulator and real IBM quantum hardware.
Abstract Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE—the heat equation in a rectangular region—and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension d≥2 d≥2 , there is an at most quadratic quantum speedup in terms of the allowable error ϵ ϵ using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative approach based on a quantum algorithm for linear equations is never faster than the best classical algorithms.
Quantum computing has the potential to transform computational problem-solving by leveraging quantum mechanical principles of superposition and entanglement. This capability is particularly important for the numerical solution of complex and/or multidimensional partial differential equations (PDEs). There are currently many quantum techniques available for solving PDEs, which are mainly based on variational quantum algorithms (VQAs). However, the existing quantum PDE solvers, particularly those based on VQAs suffer from limitations such as low accuracy, high execution times, and low scalability on both simulators as well as on real quantum devices. 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 our proposed algorithm by solving several practically useful PDEs such as Poisson, heat, Black-Scholes, and Navier-Stokes equations. Our results demonstrate higher accuracy, higher scalability, and faster execution times compared to VQA-based solvers on noise-free and noisy quantum simulators from IBM and achieved promising results on real quantum hardware.
Stochastic diffusion is a general phenomenon observed in various national and engineering systems. It is typically modeled by either stochastic differential equation (SDE) or Fokker-Planck equation (FPE), which are equivalent approaches. Path integral is an accurate and effective method to solve FPEs. Yet, computational efficiency is the common challenge for path integral and other numerical methods, include time and space complexities. Previously, one-dimensional continuous-time quantum walk was used to simulate diffusion. By combining quantum diffusion and random diffusion, the new approach can accelerate the simulation with longer time steps than those in path integral. It was demonstrated that simulation can be dozens or even hundreds of times faster. In this paper, a new generic quantum operator is proposed to simulate drift-diffusion processes in high-dimensional space, which combines quantum walks on graphs with traditional path integral approaches. Probability amplitudes are computed efficiently by spectral analysis. The efficiency of the new method is demonstrated with stochastic resonance problems.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.