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 > 0, such that log(n) depth QAOA cannot arbitrarily-well approximate the ground state energy of random diluted k-spin glasses when k ≥ 4 is even. This is equivalent to the weak approximation resistance of logarithmic depth QAOA to the Max-k-XOR 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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