The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes

TL;DR

This paper discusses quantum circuit complexity and its implications for quantum proofs, quantum money, and black hole physics, linking computational complexity with fundamental questions in quantum information and gravity.

Contribution

It unifies various topics like quantum money, black hole information paradox, and AdS/CFT through the lens of quantum circuit complexity, providing insights across multiple fields.

Findings

Quantum complexity bounds inform quantum money security

Complexity considerations shed light on black hole information paradox

Connections between quantum computation and holography are explored

Abstract

These are lecture notes from a weeklong course in quantum complexity theory taught at the Bellairs Research Institute in Barbados, February 21-25, 2016. The focus is quantum circuit complexity---i.e., the minimum number of gates needed to prepare a given quantum state or apply a given unitary transformation---as a unifying theme tying together several topics of recent interest in the field. Those topics include the power of quantum proofs and advice states; how to construct quantum money schemes secure against counterfeiting; and the role of complexity in the black-hole information paradox and the AdS/CFT correspondence (through connections made by Harlow-Hayden, Susskind, and others). The course was taught to a mixed audience of theoretical computer scientists and quantum gravity / string theorists, and starts out with a crash course on quantum information and computation in general.