Abstract
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 nqubit 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 superconstant level p = o(log log n), assuming a strengthened version of the socalled overlap gap property. This gives the first limitations on the QAOA on dense instances at superconstant level, improving upon the recent result [Basso et al., 2022].
BibTeX  Entry
@InProceedings{anshu_et_al:LIPIcs.ITCS.2023.5,
author = {Anshu, Anurag and Metger, Tony},
title = {{Concentration Bounds for Quantum States and Limitations on the QAOA from Polynomial Approximations}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {5:15:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772631},
ISSN = {18688969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17508},
URN = {urn:nbn:de:0030drops175085},
doi = {10.4230/LIPIcs.ITCS.2023.5},
annote = {Keywords: quantum computing, polynomial approximation, quantum optimization algorithm, QAOA, overlap gap property}
}
Keywords: 

quantum computing, polynomial approximation, quantum optimization algorithm, QAOA, overlap gap property 
Collection: 

14th Innovations in Theoretical Computer Science Conference (ITCS 2023) 
Issue Date: 

2023 
Date of publication: 

01.02.2023 