Quantum computers promise to solve certain problems faster than what is known classically, with prominent applications in variety of fields such as condensed matter physics or materials science. Despite advances in algorithm design, the implementation of quantum algorithms remains a major challenge due to the hardware limitations of current and near-term quantum computers. In this proposal, we develop techniques that enable the resource-efficient implementation of quantum algorithms with the overall goal of developing a more unified and streamlined quantum algorithm implementation stack.

First, we introduce Hamiltonian embedding, a technique that maps sparse matrices to larger, more structured Hamiltonians which can be more easily simulated using native device operations. Through real-machine experiments and resource analyses, we demonstrate the practicality of this approach for a variety of computational problems. Next, we develop a framework for simulating high-dimensional differential equations using Hamiltonian simulation as the main computational primitive. When utilizing Hamiltonian embedding, our approach avoids sophisticated oracle constructions and yields significant reductions in circuit depths, thereby making quantum differential equation solvers more practical for intermediate and early-fault tolerant devices. Finally, we discuss future work on the implementation of more general matrix functions and tensor network-based optimization of dynamic circuits.