Quantum computing - in which we use the unusual properties of very small particles or electronic circuits to process information - has the potential to revolutionise high-performance computing as applied across major industry sectors and branches of science. The computational capability of a quantum computer can grow exponentially, so that adding just one quantum bit will double the potential capacity. However, there are important challenges to realising the potential of these devices. These challenges are not only around building the hardware for quantum computing, but also how to programme a quantum computer in order to take advantage of the new opportunities it could offer for a particular calculation. In this project, we explore new techniques for programming quantum computers, both relevant for near-term devices that require noise mitigation and hardware-specific algorithms, and future error-corrected quantum computers. We will begin by developing new techniques to build specific quantum states by changing the parameters of the system time-dependently without adding excess energy to the system (which we refer to as optimised counterdiabatic driving). In addition, we will develop quantum algorithms for specific applications, identifying opportunities for speeding up calculations in computational fluid dynamics, plasma dynamics, or quantum science, and understanding where these might exhibit an advantage over existing conventional algorithms on supercomputers. Finally, we will test implementations of these techniques on current hardware, alongside developing techniques to verify the output of the quantum computer.