In this work, we study the limitations of the Quantum Approximate Optimization Algorithm (QAOA) through the lens of statistical physics and show that there exists $epsilon > 0$, such that $epsilonlog(n)$ depth QAOA cannot arbitrarily-well approximate the ground state energy of random diluted $k$-spin glasses when $kgeq4$ is even. This is equivalent to the weak approximation resistance of logarithmic depth QAOA to the kxors problem. We further extend the limitation to other boolean constraint satisfaction problems as long as the problem satisfies a combinatorial property called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. As a consequence of our techniques, we 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. Our results provide a new way to study the power and limit of QAOA through statistical physics methods and combinatorial properties.
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