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# Classical Algorithms and Quantum Limitations for Maximum Cut on High-Girth Graphs

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## Acknowledgements

We thank Beatrice Nash for collaborating on early stages of this project. KM thanks Jeffrey M. Epstein and Matt Hastings for explaining details on Lieb-Robinson bounds. Matt Hastings proposed a question similar to this one. Ruslan Shaydulin offered suggestions on simulating QAOA. We thank Madelyn Cain, Eddie Farhi, Pravesh Kothari, Sam Hopkins, and Leo Zhou for useful discussions. Special thanks to Leo Zhou for sharing with us the code and data from the paper [Zhou et al., 2020].

## Cite As

Boaz Barak and Kunal Marwaha. Classical Algorithms and Quantum Limitations for Maximum Cut on High-Girth Graphs. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.14

## Abstract

We study the performance of local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and their relationship to that of randomized classical algorithms. 1) We prove that every (quantum or classical) one-local algorithm (where the value of a vertex only depends on its and its neighbors' state) achieves on D-regular graphs of girth > 5 a maximum cut of at most 1/2 + C/√D for C = 1/√2 ≈ 0.7071. This is the first such result showing that one-local algorithms achieve a value that is bounded away from the true optimum for random graphs, which is 1/2 + P_*/√D + o(1/√D) for P_* ≈ 0.7632 [Dembo et al., 2017]. 2) We show that there is a classical k-local algorithm that achieves a value of 1/2 + C/√D - O(1/√k) for D-regular graphs of girth > 2k+1, where C = 2/π ≈ 0.6366. This is an algorithmic version of the existential bound of [Lyons, 2017] and is related to the algorithm of [Aizenman et al., 1987] (ALR) for the Sherrington-Kirkpatrick model. This bound is better than that achieved by the one-local and two-local versions of QAOA on high-girth graphs [M. B. Hastings, 2019; Marwaha, 2021]. 3) Through computational experiments, we give evidence that the ALR algorithm achieves better performance than constant-locality QAOA for random D-regular graphs, as well as other natural instances, including graphs that do have short cycles. While our theoretical bounds require the locality and girth assumptions, our experimental work suggests that it could be possible to extend them beyond these constraints. This points at the tantalizing possibility that O(1)-local quantum maximum-cut algorithms might be pointwise dominated by polynomial-time classical algorithms, in the sense that there is a classical algorithm outputting cuts of equal or better quality on every possible instance. This is in contrast to the evidence that polynomial-time algorithms cannot simulate the probability distributions induced by local quantum algorithms.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
##### Keywords
• approximation algorithms
• QAOA
• maximum cut
• local distributions

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