Quantum algorithms for simulating quantum systems

Conventional computers are invaluable tools for analysing and predicting the behaviour of the world around us; from the formation of the Universe at large, to the motion of subatomic particles. Unfortunately, the latter problem has proved exceedingly difficult to simulate accurately, due to the computational complexity allowed by the laws of quantum mechanics. There have been a number of proposals to `fight fire with fire' -- to use quantum computers to efficiently simulate quantum systems of interest. State-of-the-art resource estimates suggest that it may be possible to simulate classically intractable systems using hundreds of thousands, or millions, of physical qubits. Unfortunately, such resources are beyond our current capabilities -- and may remain so for the foreseeable future. This thesis presents a number of approaches to lessen the burden of quantum simulation. I introduce a hybrid quantum-classical algorithm for ansatz-based imaginary time evolution, and show how it can be used to find the ground states of Hermitian and non-Hermitian Hamiltonians of chemical relevance. While such hybrid algorithms have become popular in recent years, it is still unclear if they will be able to demonstrate quantum advantage, without the protection of quantum error correction. To partially address this question, I present a technique for error mitigation in chemical simulations, that can be used to reduce the effects of noise, at a modest cost (compared to quantum error correction). The majority of studies to date have focused on finding the electronic ground states of molecular systems. It may be prudent to investigate alternative simulation targets, which may turn out to require fewer resources. In this thesis I show how two alternative physical phenomena can be simulated on digital quantum computers: the vibrations of molecules, and muon spectroscopy experiments. For the case of molecular vibrations, I show how to map the problem onto a digital quantum computer, and extract properties of physical interest. For the case of muon spectroscopy, I present a quantum algorithm to analyse the data arising from muon spectroscopy experiments, and use numerical simulations to infer the error corrected resources required to simulate classically challenging system sizes.