In this paper, we give a one-pass quantum streaming algorithm for Max-$k$SAT that uses $\operatorname{polylog}(n)$ space and achieves a $0.7172$-approximation on instances with $n$ variables. In contrast, prior work by Chou, Golovnev, and Velusamy (FOCS 2020) implies that achieving an approximation ratio better than $\sqrt{2}/2 \approx 0.7071$ for Max-$k$SAT requires $Ω(\sqrt{n})$ space for any classical streaming algorithm. Therefore, it yields an exponential quantum space advantage for Max-$k$SAT in the streaming setting. We further give a one-pass quantum streaming algorithm for Max-2OR that uses $\operatorname{polylog}(n)$ space and achieves a $0.7425$-approximation on instances with $n$ variables. Combining with the known results, it gives a complete classification of quantum space advantages for all Boolean Max-2CSPs.

翻译：本文提出了一个用于Max-$k$SAT的单程量子流式算法，该算法使用$\operatorname{polylog}(n)$空间，在具有$n$个变量的实例上实现了0.7172的近似比。相比之下，Chou、Golovnev和Velusamy的先前工作（FOCS 2020）表明，对于任何经典流式算法，要获得优于$\sqrt{2}/2 \approx 0.7071$的Max-$k$SAT近似比，需要$\Omega(\sqrt{n})$的空间。因此，这为流式设置中的Max-$k$SAT带来了指数级的量子空间优势。我们进一步给出了一个用于Max-2OR的单程量子流式算法，该算法使用$\operatorname{polylog}(n)$空间，在具有$n$个变量的实例上实现了0.7425的近似比。结合已知结果，这为所有布尔Max-2CSP问题提供了量子空间优势的完整分类。