The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose
algorithm for combinatorial optimization problems whose performance can only
improve with the number of layers p. While QAOA holds promise as an algorithm
that can be run on near-term quantum computers, its computational power has not
been fully explored. In this work, we study the QAOA applied to the
Sherrington-Kirkpatrick (SK) model, which can be understood as energy
minimization of n spins with all-to-all random signed couplings. There is a
recent classical algorithm by Montanari that, assuming a widely believed
conjecture, can be tailored to efficiently find an approximate solution for a
typical instance of the SK model to within (1βΟ΅) times the ground
state energy. We hope to match its performance with the QAOA. Our main result
is a novel technique that allows us to evaluate the typical-instance energy of
the QAOA applied to the SK model. We produce a formula for the expected value
of the energy, as a function of the 2p QAOA parameters, in the infinite size
limit that can be evaluated on a computer with O(16p) complexity. We
evaluate the formula up to p=12, and find that the QAOA at p=11 outperforms
the standard semidefinite programming algorithm. Moreover, we show
concentration: With probability tending to one as nββ, measurements of
the QAOA will produce strings whose energies concentrate at our calculated
value. As an algorithm running on a quantum computer, there is no need to
search for optimal parameters on an instance-by-instance basis since we can
determine them in advance. What we have here is a new framework for analyzing
the QAOA, and our techniques can be of broad interest for evaluating its
performance on more general problems where classical algorithms may fail.Comment: 32 pages, 2 figures, 2 tables; improved presentation of figures and
iterative formul