UC San Diego

Quantum mechanics is one of the most successful and striking theories in physics. It predicts atomic particles can have exotic properties, such as quantum entanglement, that any classical local theory cannot describe. This phenomenon dramatically increases the complexity of nature, and it indicates there is no classical algorithm that can universally simulate all quantum many-body states. On the other hand, as opposed to classical systems, we never observe quantum properties directly since the measurement for the quantum systems is destructive. People can only determine the quantum black box by the statistics of classical readouts. The complexity of quantum objects implies exponentially many measurements and classical data to figure out the quantum states fully. It is underlying those challenges to find an efficient classical representation of quantum many-body states. An efficient (classical) representation will require fewer classical data of quantum states and learn many of its properties. And an efficient representation can also be served as a classical simulation algorithm for the quantum states. The is no universal, efficient representation for all the quantum states, and it usually depends on the learning properties or underlying quantum states. This thesis will give two efficient representations: the classical shadow representation of quantum states and the hierarchical representation of quantum states. We will see that those efficient representations will help us learn and simulate quantum many-body states and lead to many critical applications in quantum information technology, condensed matter physics, and quantum field theory.