Quantum proofs (Q2808277)

Quantum interactive proof systems use quantum information, interaction and proofs. The superposition principle appears to offer computational advantage over classical proof strings. In the interactive proof system setting, one can consider one verifier and several provers that exchange and process quantum information, giving rise to quantum complexity classes. NEWLINENEWLINENEWLINE The monograph under review is a survey of many known results concerning quantum proofs. Classical interactive proof systems depart from non-interactive settings when the verifier makes use of randomness, satisfied with statistical evidence. NEWLINENEWLINENEWLINE Chapter two follows the introductory Chapter one and deals with some preliminary material of quantum computing and complexity theory. NEWLINENEWLINENEWLINE Chapter three introduces the class QMA of languages that have efficiently verifiable quantum proofs. In QMA are the decision problems whose positive instances have quantum proofs that can be verified by an efficient quantum procedure, it satisfies many features of NP. The class QIP corresponds to single-prover quantum interactive proofs. They can be parallelised to three message interactions by using the superposition principle. Zero-interactions proofs are related to the non-cloning theorem. NEWLINENEWLINENEWLINE Chapter four deals with single-prover quantum interactive proof systems. NEWLINENEWLINENEWLINE Chapter five describes the class QSZK of quantum zero-knowledge interactive proofs. The difference to the classical systems is indicated. For quantum systems it is difficult to extend key techniques such as rewinding. Known quantum analogues are presented. NEWLINENEWLINENEWLINE Chapter six deals with multi-prover interactive proofs. The entanglement of different provers is investigated. This entanglement goes against the usual intuition on which classical systems are built. NEWLINENEWLINENEWLINE The survey is intended for non-specialists, however, the reader is required to have some basic knowledge in quantum computing as well as complexity theory. Full proofs of the main results are not included, instead detailed sketches highlight the ideas behind the proofs to simplify the reading. Each chapter ends with notes and references for the discussed results, a brief survey of related results and some indications to the literature. An index is not included. The monograph represents an up-to-date survey of the domain of quantum proofs and is recommended to anyone who is interested in this topic.