The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size

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 that be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study applied to Sherrington-Kirkpatrick (SK) model, which understood energy minimization xmlns:mml="http://www.w3.org/1998/Math/MathML">n spins all-to-all random signed couplings. There recent classical by Montanari that, assuming widely believed conjecture, efficiently find approximate solution typical instance SK model within xmlns:mml="http://www.w3.org/1998/Math/MathML">(1−ϵ) times ground state energy. We hope match QAOA.Our main result novel technique allows us evaluate typical-instance model. produce formula expected value energy, function xmlns:mml="http://www.w3.org/1998/Math/MathML">2p parameters, in infinite size limit evaluated computer xmlns:mml="http://www.w3.org/1998/Math/MathML">O(16pp=12, and at xmlns:mml="http://www.w3.org/1998/Math/MathML">p=11 outperforms standard semidefinite programming algorithm. Moreover, show concentration: With probability tending one xmlns:mml="http://www.w3.org/1998/Math/MathML">n→∞, measurements will strings energies concentrate our calculated value. As running computer, there no need search optimal parameters instance-by-instance basis since determine them advance. What have here new framework analyzing QAOA, techniques broad interest evaluating more general where algorithms may fail.