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.