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 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.
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.
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.
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.
The framework of this thesis is fault-tolerant quantum algorithms. Grover's algorithm and quantum walks are described in Chapter 2. We start by highlighting the central role that rotations play in quantum algorithms, explaining Grover's, why it is optimal, and how it may be extended. Key subroutines explained in this area are amplitude amplification and quantum walks, which will constitute useful parts of other algorithms. In the third chapter, in contrast, we turn to the exponential advantages promised by the Fourier transform in the context of the hidden subgroup problem. However, since this application is restricted to cryptography, we later explore its use in quantum linear algebra problems. Here we explain the development of the original quantum linear solver algorithm, its improvements, and finally the dequantization techniques that would often restrict the quantum advantage to polynomial. Chapter 4 is concerned with quantum simulation. We will review classical quantum chemistry techniques, and then focus on Hamiltonian simulation and ground state preparation as the key problems to be solved. Hamiltonian simulation, in particular, will enable the use of quantum phase estimation which computes the eigenvalues or energies of a given quantum state. Given the tradition of our group with error correction, we could not end this thesis without dedicating a final chapter to this topic. Here we explain the most important quantum error correction codes, the surface and color codes, and one extension of the latter, gauge color codes. They will show the complexity of implementing non-Clifford quantum gates, therefore validating their consideration as the bottleneck metric.
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.
Quantum computing offers promising alternatives to classical approaches for solving complex linear algebra problems. This paper presents a comparative study of the performance of quantum algorithms versus classical algorithms in solving systems of linear equations and matrix operations. Through simulation and analysis, we demonstrate that while quantum computing holds advantages in specific problem sets, classical computing remains efficient for general applications. These findings highlight the current limitations and potential of quantum computing.