CCF: AF: Small: Simulating Hamiltonian dynamics: Algorithms and applications

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Simulating the dynamics of quantum systems is a major potential application of quantum computers. Quantum simulation can be applied, for example, to model chemical reactions and to predict the behavior of materials. It can also be used as a tool to develop other quantum algorithms, and its study sheds light on the power of quantum computers. This project focuses on two related aspects of quantum simulation: developing improved quantum algorithms for basic simulation problems and studying applications of quantum simulation to other computational tasks.While the state of the art in quantum simulation algorithms is fairly advanced, the ultimate limits for this fundamental task remain unclear. Operating within a broadly applicable framework of quantum systems described by sparse matrices, the project aims to develop optimal tradeoffs among all relevant simulation parameters. It also studies quantum algorithms for simulating open quantum systems, leveraging algorithmic ideas from the well-studied scenario of noiseless systems to give new algorithms for the more realistic setting of noisy quantum systems that interact with their environments. Furthermore, the project studies applications of quantum simulation algorithms. Quantum simulation is the core component of a quantum algorithm for extracting information about the solutions of large systems of linear equations. The project applies recent improvements to quantum simulation algorithms to give improved quantum algorithms for linear systems. It also studies the computational power of particular types of linear systems, such as those that arise in the computation of effective resistances in electrical networks. Finally, the project investigates how algorithmic ideas from advanced quantum simulation algorithms can be applied outside the context of simulation, e.g., in computational number theory.