A Quantum Unique Games Conjecture

TL;DR

This work defines quantum extensions of Label-Cover and Unique-Label-Cover and introduces the quantum value framework for CSPs, including the Quantum Unique Games Conjecture (qUGC) and its variants. It shows how classical reductions from UL-Cover can be quantized to derive hardness for quantum CSPs, establishing RE-hardness results for 2-Lin_q and, under wqUGC, for MaxCut_q, while highlighting an integrality-gap phenomenon between SDP and quantum relaxations. The authors develop the Projectivization and Folding Lemmas as central technical tools to translate between quantum and classical reductions, enabling robust hardness transfers across CSPs and connecting to the MIP* = RE landscape. The findings imply that, under the conjectured quantum hardness assumptions, the Goemans-Williamson bound remains essentially optimal for quantum MaxCut, and they illuminate a broader program toward a quantum hardness-of-approximation theory for noncommutative CSPs with potential applications to QMA and quantum Hamiltonian complexity.

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.

Figures (12)

• Figure 1: An instance of Max-Cut and an example of a partition (cut).
• Figure 2: Trivial Reductions: if a problem in this diagram is hard (for some complexity class), then every problem reachable from it is at least as hard.
• Figure 3: Implication of the PCP theorem: problems in the third column are $\mathrm{NP}$-hard.
• Figure 4: The complexity landscape of Label-Cover.
• Figure 5: The complexity landscape of Unique-Label-Cover assuming both UGC and qUGC.

Theorems & Definitions (54)

• Theorem : PCP Theorem
• Theorem : $\mathrm{MIP}^*=\mathrm{RE}$, informal
• Conjecture : UGC khot_original, informal
• Conjecture : qUGC, informal
• Theorem
• Theorem : Theorem \ref{['thm:maxcuthardness']}, informal
• Theorem : KKMO, informal
• Definition 1: Simultaneous measurability
• Definition 2: Self-commuting measurements
• Definition 3