Quantum Algorithms for PDEs via Summation-by-Parts Discretisations

Abstract

Partial differential equations (PDEs) underpin the mathematical description of physical phenomena across science and engineering. High-order discretisation techniques, such as those based on the Summation-by-Parts (SBP) framework, provide accurate and energy-stable numerical schemes that preserve conservation and stability properties of the continuous equations. These discretisations yield large, linear systems that have significant structure, whose solution constitutes the computational bottleneck of many scientific simulations. Quantum systems can represent exponentially large state spaces using only a polynomial number of qubits. By exploiting this representational capacity, quantum algorithms can achieve substantial, and in some cases exponential, speedups for certain classes of computational problems. However, the practical application of quantum algorithms to PDEs is challenging as a result of the difficulty of efficiently representing the discrete operators as unitary transformations suitable for quantum computation.

This thesis establishes a first step and a significant step toward a unified framework linking high-order SBP discretisations with quantum algorithms based on polynomial spectral transformations. Using Quantum Singular Value Transformation (QSVT), we show how the matrix exponential exp(AT), governing time evolution in semi-discrete PDE systems, can be efficiently implemented as a quantum circuit. A key contribution is the development of systematically constructible and ancilla-efficient block-encodings for structured matrices, including those arising from high-order SBP discretisations of PDEs. These constructions exploit the tensor-product and sparsity structure of the underlying differential operators to enable automated coherent circuit synthesis with polylogarithmic scaling in system size. The resulting framework bridges classical numerical methods and quantum algorithm design by embedding stable, high-order discretisation operators into the coherent quantum model.

The methods are demonstrated on the linear advection equation by block-encoding the SBP semi-discrete operator and applying QSVT to realise its time evolution. This work serves as a blueprint for applying QSVT-based quantum algorithms to high-order discretisations of general linear PDEs.