Abstract The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the multidimensional transport equation with space- and time-dependent coefficients. We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement of relevant observables. The evolution step combines a high-order centered finite difference with a time-splitting scheme based on product formula approximations, also known as Trotterization. We introduce a novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits, compared with previously established operator-norm analysis. This new scaling significantly reduces the projected computational resources, independently of the circuit implementation of the trotterized evolution operator. We also present efficient quantum circuits based on (sparse) Walsh approximations along with numerical simulations that confirm the predicted vector-norm scaling. We report results on real quantum hardware for the one-dimensional convection equation, and solve a non-linear ordinary differential equation via its associated Liouville equation, a particular case of transport equations. This work provides a practical framework for efficiently simulating transport phenomena on quantum computers, with potential applications in plasma physics, molecular gas dynamics and non-linear dynamical systems, including chaotic systems.
Abstract The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the multidimensional transport equation with space- and time-dependent coefficients. We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement of relevant observables. The evolution step combines a high-order centered finite difference with a time-splitting scheme based on product formula approximations, also known as Trotterization. We introduce a novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits, compared with previously established operator-norm analysis. This new scaling significantly reduces the projected computational resources, independently of the circuit implementation of the trotterized evolution operator. We also present efficient quantum circuits based on (sparse) Walsh approximations along with numerical simulations that confirm the predicted vector-norm scaling. We report results on real quantum hardware for the one-dimensional convection equation, and solve a non-linear ordinary differential equation via its associated Liouville equation, a particular case of transport equations. This work provides a practical framework for efficiently simulating transport phenomena on quantum computers, with potential applications in plasma physics, molecular gas dynamics and non-linear dynamical systems, including chaotic systems.
In this work, we tackle the resolution of partial differential equations (PDEs) on digital quantum computers. Two fundamental PDEs are addressed: the anisotropic diffusion equation and the anisotropic convection equation. We present a quantum numerical scheme consisting of three steps: quantum state preparation, evolution with diagonal operators, and measurement of observables of interest. The evolution step relies on a high-order centered finite difference and a product formula approximation, also known as Trotterization. We provide novel vector-norm analysis to bound the different sources of error. We prove that the number of time-steps required in the evolution can be reduced by a factor Θ(16n) for the diffusion equation, and Θ(4n) for the convection equation, where n is the number of qubits per dimension, an exponential reduction compared to the previously established operator-norm analysis.
Current classical simulations of physical phenomena face many limitations, particularly when dealing with complex systems such as turbulent fluids, plasmas or strongly correlated quantum materials. These simulations often struggle with high computational costs, limited scalability across multiple dimensions and time scales, and difficulties in capturing inherently non-linear or chaotic behaviors. Even with access to powerful high performance computing (HPC) resources, accurately resolving fine-scale dynamics or long time evolutions remains a significant challenge. The development of novel numerical methods is therefore essential for both academic research and industrial applications. Quantum computers emerge as novel numerical tools for studying such systems and solving the partial differential equations that model them. Thanks to the superposition and entanglement properties of qubits, digital computations can be performed in novel ways, offering promising theoretical speedups for certain problems. In this context, this thesis contributes to the development and understanding of quantum algorithms, ranging from the design of quantum circuits to the numerical solutions of partial and ordinary differential equations. First, efficient quantum circuits are presented for the implementation of crucial routines that are prevalent in quantum algorithms (multi-controlled operations, quantum state preparation, diagonal operations), and for a novel routine: the quantum Laplace transform. These routines limited the efficiency of quantum algorithms due to generic implementations that required complexities scaling exponentially with the number of qubits. Now, these tasks can be performed with tailored quantum circuits that leverage the structure of the physical problems. In the second part, quantum numerical schemes are introduced to address anisotropic transport equations and non-linear Hamiltonian ODEs. These schemes are composed of three steps (initialization, evolution, and measurement) and are based on high-ordered finite-difference and product formula approximations, also known as Trotterization. Efficient protocols to measure quantities of interest are presented, and a novel numerical analysis based on the vector norm instead of the generic operator norm is introduced, proving that the number of time-steps can be reduced by a factor that scales exponentially with the number of qubits while reaching similar accuracy. Additionally, non-linear Madelung transforms are studied to solve fluid equations on quantum computers. The various results of this thesis are illustrated with classical simulations of the quantum algorithms and simulations on real quantum hardware. This work lays the foundations for the practical simulations of classical physical systems on both current noisy-intermediate-scale quantum devices and future fault-tolerant quantum computers.
Transport phenomena play a key role in a variety of application domains, and efficient simulation of these dynamics remains an outstanding challenge. While quantum computers offer potential for significant speedups, existing algorithms either lack rigorous theoretical guarantees or demand substantial quantum resources, preventing scalable and efficient validation on realistic quantum hardware. To address this gap, we develop a comprehensive framework for simulating classes of transport equations, offering both rigorous theoretical guarantees -- including exponential speedups in specific cases -- and a systematic, hardware-efficient implementation. Central to our approach is the Hamiltonian embedding technique, a white-box approach for end-to-end simulation of sparse Hamiltonians that avoids abstract query models and retains near-optimal asymptotic complexity. Empirical resource estimates indicate that our approach can yield an order-of-magnitude (e.g., 42×) reduction in circuit depth given favorable problem structures. We then apply our framework to solve linear and nonlinear transport PDEs, including the first experimental demonstration of a 2D advection equation on a trapped-ion quantum computer.
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.
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.
The Vlasov--Poisson system provides the fundamental kinetic description of plasma and plays a central role in understanding collective phenomena such as Landau damping and wave--particle interactions. Efficient numerical simulation of these dynamics remains challenging because of the high dimensionality of phase space. In this work, we studied magnetized and non-magnetized plasma using a quantum simulation framework for the linearized Vlasov-Poisson equation by reformulating the discretized system as a Hermitian Hamiltonian suitable for gate-based quantum computation. The time evolution is implemented using first, second and fourth-order Trotter--Suzuki product formulas and the recently proposed Time-Resolved Interaction Framework (THRIFT). The performance of the different simulation methods is systematically evaluated through electric field evolution, state fidelity, convergence behavior, energy conservation, entanglement entropy and quantum resource requirements, including circuit depth and two-qubit gate complexity, for both magnetized and non-magnetized plasma models. To further reduce finite time step errors without increasing circuit depth, Richardson extrapolation is incorporated as a error-mitigation technique. The results provide a comprehensive comparison of Trotter and THRIFT approaches and establish practical guidelines for accurate and resource-efficient quantum simulation of plasma dynamics on gate-based quantum computers.