The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut
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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 - ε)-approximation requires Ω(n) space (a 2-approximation is trivial with O(log n) space). We generalize both of these qualifiers, demonstrating Ω(n) space lower bounds for (2 - ε)-approximating Max-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 + ε)-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.
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