While the search for quantum advantage typically focuses on speedups in execution time, quantum algorithms also offer the potential for advantage in space complexity. Previous work has shown such advantages for data stream problems, in which elements arrive and must be processed sequentially without random access, but these have been restricted to specially-constructed problems [Le Gall, SPAA `06] or polynomial advantage [Kallaugher, FOCS `21]. We show an exponential quantum space advantage for the maximum directed cut problem. This is the first known exponential quantum space advantage for any natural streaming problem. This also constitutes the first unconditional exponential quantum resource advantage for approximating a discrete optimization problem in any setting. Our quantum streaming algorithm $0.4844$-approximates the value of the largest directed cut in a graph stream with $n$ vertices using polylog$(n)$ space, while previous work by Chou, Golovnev, and Velusamy [FOCS '20] implies that obtaining an approximation ratio better than $4/9 \approx 0.4444$ requires $\Omega(\sqrt{n})$ space for any classical streaming algorithm. Our result is based on a recent $\widetilde{\text{O}}(\sqrt{n})$ space classical streaming approach by Saxena, Singer, Sudan, and Velusamy [FOCS '23], with an additional improvement in the approximation ratio due to recent work by Singer [APPROX '23].

翻译：尽管量子优势的探索通常聚焦于执行时间上的加速，量子算法在空间复杂度方面同样具备潜在优势。已有研究展示了针对数据流问题的此类优势（数据流中元素需顺序处理且无法随机访问），但这些成果局限于特殊构造问题[Le Gall, SPAA '06]或多项式级优势[Kallaugher, FOCS '21]。我们证明了最大有向割问题存在指数级量子空间优势，这是首个已知自然流式问题中指数级量子空间优势的实例，同时也构成了任何设定下离散优化问题近似求解中无条件指数级量子资源优势的首个案例。我们的量子流式算法以polylog(n)空间实现了对含n个顶点的图流中最大有向割值的0.4844-近似；而Chou、Golovnev和Velusamy [FOCS '20]的前期工作表明，对于任何经典流式算法，若要获得优于4/9≈0.4444的近似比，则需Ω(√n)空间。本结果基于Saxena、Singer、Sudan和Velusamy [FOCS '23]近期提出的经典流式方法（空间复杂度为Õ(√n)），并借助Singer [APPROX '23]的最新工作在近似比上实现了进一步优化。