The complexity of free games with two or more classical players was essentially settled by Aaronson, Impagliazzo, and Moshkovitz (CCC'14). There are two complexity classes that can be considered quantum analogues of classical free games: (1) AM*, the multiprover interactive proof class corresponding to free games with entangled players, and, somewhat less obviously, (2) BellQMA(2), the class of quantum Merlin-Arthur proof systems with two unentangled Merlins, whose proof states are separately measured by Arthur. In this work, we make significant progress towards a tight characterization of both of these classes. 1. We show a BellQMA(2) protocol for 3SAT on $n$ variables, where the total amount of communication is $\tilde{O}(\sqrt{n})$. This answers an open question of Chen and Drucker (2010) and also shows, conditional on ETH, that the algorithm of Brand\~{a}o, Christandl and Yard (STOC'11) is tight up to logarithmic factors. 2. We show that $\mathsf{AM}^*[n_{\text{provers}} = 2, q = O(1), a =\mathrm{poly}\log(n)] = \mathsf{RE}$, i.e. that free entangled games with constant-sized questions are as powerful as general entangled games. Our result is a significant improvement over the headline result of Ji et al. (2020), whose MIP* protocol for the halting problem has $\mathrm{poly}(n)$-sized questions and answers. 3. We obtain a zero-gap AM* protocol for a $\Pi_2$ complete language with constant-size questions and almost logarithmically large answers, improving on the headline result of Mousavi, Nezhadi and Yuen (STOC'22). 4. Using a connection to the nonuniform complexity of the halting problem we show that any MIP* protocol for RE requires $\Omega(\log n)$ bits of communication. It follows that our results in item 3 are optimal up to an $O(\log^* n)$ factor, and that the gapless compression theorems of MNY'22 are asymptotically optimal.
翻译:与两个或更多经典玩家的自由游戏的复杂性基本上由 Aaronson、 Impagliazzo 和 Moshkovitz (CCC'14) 解决。 有两个复杂的游戏类别可以被视为经典自由游戏的量级类比:(1) AM*, 与玩家交织的免费游戏相对应的多倍互动证明类比, 更不明显地, (2) BellQMA(2), 由两个不连结的 Merlin( Merhur) 类量的量子 Merlin- Arthur 校验系统, 其证明状态由Arthur单独测量 。 在这项工作中, 我们为这两个类的严格描述。 1, 我们为 3 美元 的 3 美元, 3 美元 美元, 美元 美元= 美元= 美元 协议的数值。 这回答了Chen 和 Druck ( ) 的开放性问题, 取决于 ET, Brandái, i, i, Christandl and Yard (ST'11) 的算算算算算得 。