Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complexity lower bounds for computing the approximately-commuting
  operator value of non-local games to high precision   We study the problem of approximating the commuting-operator value of a
two-player non-local game. It is well-known that it is $\mathrm{NP}$-complete
to decide whether the classical value of a non-local game is 1 or $1-
\epsilon$. Furthermore, as long as $\epsilon$ is small enough, this result does
not depend on the gap $\epsilon$. In contrast, a recent result of Fitzsimons,
Ji, Vidick, and Yuen shows that the complexity of computing the quantum value
grows without bound as the gap $\epsilon$ decreases. In this paper, we show
that this also holds for the commuting-operator value of a game. Specifically,
in the language of multi-prover interactive proofs, we show that the power of
$\mathrm{MIP}^{co}(2,1,1,s)$ (proofs with two provers, one round, completeness
probability $1$, soundness probability $s$, and commuting-operator strategies)
can increase without bound as the gap $1-s$ gets arbitrarily small.
  Our results also extend naturally in two ways, to perfect zero-knowledge
protocols, and to lower bounds on the complexity of computing the
approximately-commuting value of a game. Thus we get lower bounds on the
complexity class $\mathrm{PZK}$-$\mathrm{MIP}^{co}_{\delta}(2,1,1,s)$ of
perfect zero-knowledge multi-prover proofs with approximately-commuting
operator strategies, as the gap $1-s$ gets arbitrarily small. While we do not
know any computable time upper bound on the class $\mathrm{MIP}^{co}$, a result
of the first author and Vidick shows that for $s = 1-1/\text{poly}(f(n))$ and
$\delta = 1/\text{poly}(f(n))$, the class $\mathrm{MIP}^{co}_\delta(2,1,1,s)$,
with constant communication from the provers, is contained in
$\mathrm{TIME}(\exp(\text{poly}(f(n))))$. We give a lower bound of
$\mathrm{coNTIME}(f(n))$ (ignoring constants inside the function) for this
class, which is tight up to polynomial factors assuming the exponential time
hypothesis.

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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: math
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: stat