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.
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.
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.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian. We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits. The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters. The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms. We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cases that uses QLSAs for state preparation and post-processing.
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.
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.
Recent advances in quantum hardware and quantum computing algorithms promise significant breakthroughs in computational capabilities. Quantum computers can achieve exponential improvements in speed vs classical computers by employing principles of quantum mechanics like superposition and entanglement. However, designing quantum algorithms to solve the nonlinear partial differential equations governing fluid dynamics is challenging due to the inherent linearity of quantum mechanics, which requires unitary transformation. In this study, we first address in detail several challenges that arise when trying to deal with nonlinearity using quantum algorithms and then propose a novel pure quantum algorithm for solving a nonlinear Burgers' equation. We employed multiple copies of the state vector to calculate the nonlinear term, which is necessary due to the no-cloning theorem. By reusing qubits from the previous time steps, we significantly reduced the number of qubits required for multi-step simulations, from exponential/quadratic scaling in earlier studies to linear scaling in time in the current study. We also employed various advanced quantum techniques, including block-encoding, quantum Hadamard product, and the linear combination of unitaries, to design a quantum circuit for the proposed quantum algorithm. The quantum circuit was executed on quantum simulators, and the obtained results demonstrated excellent agreement with those from classical simulations.