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

TL;DR

This paper establishes space complexity lower bounds for approximating Max-Cut and Quantum Max-Cut in streaming models, and provides an efficient algorithm for Quantum Max-Cut, advancing understanding of quantum and classical graph problems.

Contribution

It proves tight space lower bounds for approximating Max-Cut and Quantum Max-Cut, and introduces Fourier analysis techniques to quantum communication complexity.

Findings

Any $(2 - ext{ε})$-approximation requires $ ext{Ω}(n)$ space.

An $ ext{O}( ext{log} n)$ space algorithm achieves a $(2 + ext{ε})$-approximation for Quantum Max-Cut.

First application of Boolean Fourier analysis to sequential one-way quantum communication.

Abstract

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 \emph{classical} problem, showing that any $(2 - \varepsilon)$-approximation requires $\Omega(n)$ space (a $2$-approximation is trivial with $\textrm{O}(\log n)$ space). We generalize both of these qualifiers, demonstrating $\Omega(n)$ space lower bounds for $(2 - \varepsilon)$-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 $(2 + \varepsilon)$-approximation to the Quantum Max-Cut value of a graph in $\textrm{O}(\log n)$ space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using $\textrm{o}(n)$ 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.