Approximation algorithms for noncommutative constraint satisfaction problems
CoRR(2023)
摘要
We study operator - or noncommutative - variants of constraint satisfaction
problems (CSPs). These higher-dimensional variants are a core topic of
investigation in quantum information, where they arise as nonlocal games and
entangled multiprover interactive proof systems (MIP*). The idea of
higher-dimensional relaxations of CSPs is also important in the classical
literature. For example since the celebrated work of Goemans and Williamson on
Max-Cut, higher dimensional vector relaxations have been central in the design
of approximation algorithms for classical CSPs.
We introduce a framework for designing approximation algorithms for
noncommutative CSPs. Prior to this work Max-2-Lin(k) was the only family of
noncommutative CSPs known to be efficiently solvable. This work is the first to
establish approximation ratios for a broader class of noncommutative CSPs.
In the study of classical CSPs, k-ary decision variables are often
represented by k-th roots of unity, which generalise to the noncommutative
setting as order-k unitary operators. In our framework, using representation
theory, we develop a way of constructing unitary solutions from SDP
relaxations, extending the pioneering work of Tsirelson on XOR games. Then, we
introduce a novel rounding scheme to transform these solutions to order-k
unitaries. Our main technical innovation here is a theorem guaranteeing that,
for any set of unitary operators, there exists a set of order-k unitaries
that closely mimics it. As an integral part of the rounding scheme, we prove a
random matrix theory result that characterises the distribution of the relative
angles between eigenvalues of random unitaries using tools from free
probability.
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