Limitations of Local Quantum Algorithms on Random MAX-k-XOR and Beyond

Authors Chi-Ning Chou, Peter J. Love, Juspreet Singh Sandhu, Jonathan Shi

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Author Details

Chi-Ning Chou
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Peter J. Love
  • Department of Physics and Astronomy, Tufts University, Medford, MA, USA
Juspreet Singh Sandhu
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Jonathan Shi
  • Department of Computing Sciences, Bocconi University, Milan, Italy


We thank Jonathan Wurtz for many insightful discussions about QAOA. We are grateful to Amartya Shankha Biswas for patiently explaining the factors of i.i.d. framework to us. We would also like to thank Antares Chen for many invigorating and profound discussions which culminated as the open problem proposed in Problem 5.1. Lastly, we would like to thank Boaz Barak for providing detailed and helpful feedback on a prior version of this manuscript, and David Gamarnik for his explanations on the state of the art results in the research area.

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Chi-Ning Chou, Peter J. Love, Juspreet Singh Sandhu, and Jonathan Shi. Limitations of Local Quantum Algorithms on Random MAX-k-XOR and Beyond. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 41:1-41:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We introduce a notion of generic local algorithm, which strictly generalizes existing frameworks of local algorithms such as factors of i.i.d. by capturing local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). Motivated by a question of Farhi et al. [arXiv:1910.08187, 2019], we then show limitations of generic local algorithms including QAOA on random instances of constraint satisfaction problems (CSPs). Specifically, we show that any generic local algorithm whose assignment to a vertex depends only on a local neighborhood with o(n) other vertices (such as the QAOA at depth less than εlog(n)) cannot arbitrarily-well approximate boolean CSPs if the problem satisfies a geometric property from statistical physics called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. We show that the random MAX-k-XOR problem has this property when k ≥ 4 is even by extending the corresponding result for diluted k-spin glasses. Our concentration lemmas confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth - in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. One of these lemmas is a strengthening of McDiarmid’s inequality, applicable when the random variables have a highly biased distribution, and may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Combinatorics
  • Theory of computation → Quantum complexity theory
  • Quantum Algorithms
  • Spin Glasses
  • Hardness of Approximation
  • Local Algorithms
  • Concentration Inequalities
  • Overlap Gap Property


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