The Quantum PCP Conjecture

TL;DR

This paper surveys the ongoing research and challenges in establishing a quantum analogue of the classical PCP theorem, exploring its implications for quantum complexity, entanglement, and quantum information theory.

Contribution

It provides a comprehensive overview of the current status, open problems, and foundational questions related to the quantum PCP conjecture.

Findings

The quantum PCP conjecture remains unresolved.

Connections between quantum entanglement and computational complexity are explored.

The survey highlights key challenges and future directions in quantum complexity theory.

Abstract

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.