A Quantum Unique Games Conjecture

TL;DR

This paper introduces quantum versions of Label-Cover and Unique-Label-Cover problems, establishing their importance in understanding the inapproximability of quantum constraint satisfaction problems, akin to classical complexity theory.

Contribution

It defines quantum extensions of classical problems and demonstrates their fundamental role in quantum inapproximability research.

Findings

Quantum Label-Cover and Unique-Label-Cover are crucial for quantum inapproximability.

These problems mirror their classical counterparts in significance.

Foundational definitions for quantum constraint satisfaction problems are provided.

Abstract

After the NP-hardness of computational problems such as 3SAT and MaxCut was established, a natural next step was to explore whether these problems remain hard to approximate. While the quantum extensions of some of these problems are known to be hard-indeed undecidable-their inapproximability remains largely unresolved. In this work, we introduce definitions for the quantum extensions of Label-Cover and Unique-Label-Cover. We show that these problems play a similarly crucial role in studying the inapproximability of quantum constraint satisfaction problems as they do in the classical setting.