Concentration Bounds for Quantum States and Limitations on the QAOA from Polynomial Approximations

Authors Anurag Anshu, Tony Metger

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Anurag Anshu
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Tony Metger
  • Institute for Theoretical Studies, ETH Zürich, Switzerland


We thank Joao Basso, David Gamarnik, Song Mei, and Leo Zhou for very helpful discussions and especially for suggesting the application to symmetric QAOA. AA also thanks Daniel Stilck França, Tomotaka Kuwahara, Cambyse Rouzé, and Juspreet Singh Sandhu for helpful discussions. This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.

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Anurag Anshu and Tony Metger. Concentration Bounds for Quantum States and Limitations on the QAOA from Polynomial Approximations. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 5:1-5:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from [De Palma et al., 2022]; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form e^{ιH^{(p)}} ⋯ e^{ιH^{(1)}} |ψ₀⟩ for any n-qubit product state |ψ₀⟩, where each H^{(i)} can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approximations to show that these states are close to local operators. This implies that the distribution of the Hamming weight of a computational basis measurement (and of other related observables) concentrates. An example of (iii) are the states produced by the quantum approximate optimisation algorithm (QAOA). Using our concentration results for these states, we show that for a random spin model, the QAOA can only succeed with negligible probability even at super-constant level p = o(log log n), assuming a strengthened version of the so-called overlap gap property. This gives the first limitations on the QAOA on dense instances at super-constant level, improving upon the recent result [Basso et al., 2022].

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Quantum complexity theory
  • quantum computing
  • polynomial approximation
  • quantum optimization algorithm
  • QAOA
  • overlap gap property


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