Approximation algorithms for noncommutative CSPs

TL;DR

This work analyzes the approximability of noncommutative constraint satisfaction problems (NC-CSPs), focusing on NC-Max-3-Cut. It develops a unified framework combining approximate isometries, relative distributions, and $\ast$-anticommutation (via generalized Weyl-Brauer operators) to achieve a $0.864$-approximation in polynomial time, with extensions to homogeneous NC-HMax-Lin$(k)$ and NC-SMax-Lin$(k)$. The analysis merges analytic and algebraic techniques, including a novel integral fidelity formula against the wrapped Cauchy relative distribution and a dimension-efficient construction, yielding strong links to Tsirelson’s results, nonlocal games, and Grothendieck-type inequalities. The results illuminate a potential complexity transition in NC-CSPs and open several directions, such as improving constants, higher-level SDP relaxations, and extending the framework to broader noncommutative problems. Overall, the paper provides new algorithmic tools for quantum-inspired CSPs and deepens connections between noncommutative optimization, probability, and operator algebra.

Abstract

Noncommutative constraint satisfaction problems (NC-CSPs) are higher-dimensional operator extensions of classical CSPs. Despite their significance in quantum information, their approximability remains largely unexplored. A notable example of a noncommutative CSP that is not solvable in polynomial time is NC-Max-$3$-Cut. We present a $0.864$-approximation algorithm for this problem. Our approach extends to a broader class of both classical and noncommutative CSPs. We introduce three key concepts: approximate isometry, relative distribution, and $\ast$-anticommutation, which may be of independent interest.

Figures (8)

• Figure 1: An assignment of $\pm 1$ to each variable $x_i$ indicates which side of the partition the corresponding vertex $i$ resides. Therefore, the expression $\tfrac{1-x_ix_j}{2}$ in the objective function is $1$ if the edge $(i,j)$ crosses the partition and zero otherwise.
• Figure 2: Transitions in complexity for classical and noncommutative $\mathrm{Max}\textit{-}\mathrm{Cut}$.
• Figure 3: $\mathrm{Max}\textit{-}{3}\textit{-}\mathrm{Cut}$ and its presentation as a polynomial optimization: Here $1,\omega,\omega^2$ are the $3$rd roots of unity and $x^*$ is the complex conjugate. The term $\tfrac{2 - x_i^* x_j - x_j^*x_i}{3}$ is $1$ if $x_i\neq x_j$ and $0$ otherwise.
• Figure 4: Conjectured transition in complexity for classical and noncommutative $\mathrm{Max}\textit{-}{3}\textit{-}\mathrm{Cut}$. $\mathsf{RE}$ is the class of problems reducible to the Halting problem.
• Figure 5: Transition in complexity for classical and noncommutative $3$-XOR. Unlike the examples of $\mathrm{Max}\textit{-}\mathrm{Cut}$ and $\mathrm{Max}\textit{-}{3}\textit{-}\mathrm{Cut}$ the transition in complexity is fully settled for both classical and noncommutative $3$-XOR.

Theorems & Definitions (109)

• Theorem 1.1: Ji, Ji et al., and Harris ji2013binaryji_mip_reharris2023universality
• Theorem 1.2
• Proposition 1.4: Isometric embedding of vectors into unitaries, Tsirelson
• Theorem 1.5: Tsirelson's Theorem
• Theorem 1.6: Theorem \ref{['thm:main-motivation']} restated for approximate isometries
• Corollary 1.7
• Remark
• Theorem 1.9: Cauchy law, informal
• Remark
• Theorem 1.10