The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut

We investigate the space complexity of two graph streaming problems: MAX-CUT and its quantum analogue, QUANTUM MAX-CUT. Previous work by Kapralov and Krachun [STOC '19] resolved the classical complexity of the classical problem, showing that any (2 - epsilon)-approximation requires Omega(n) space (a 2-approximation is trivial with O(log n) space). We generalize both of these qualifiers, demonstrating Omega(n) space lower bounds for (2-epsilon)-approximatingMAX-CUT and QUANTUM MAX-CUT, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for QUANTUM MAX-CUT only gives a 4-approximation, we show tightness with an algorithm that returns a (2 + epsilon)-approximation to the QUANTUM MAX-CUT value of a graph in O(log n) space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using o(n) space. We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.