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

Edward Farhi1,2, Jeffrey Goldstone2, Sam Gutmann, and Leo Zhou1,3

1Google Inc., Venice, CA 90291, USA
2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3Department of Physics, Harvard University, Cambridge, MA 02138, USA

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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 efficiently find an approximate solution for a typical instance of the SK model to within $(1-\epsilon)$ 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 $O(16^p)$ 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 $n\to\infty$, 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.

This work studies the performance of a general-purpose quantum algorithm for combinatorial optimization, called the QAOA, applied to the famous Sherrington-Kirkpatrick (SK) model of spin glass. This is the problem of energy minimization of all-to-all randomly coupled spins. The authors produce a formula for calculating the expected value of the energy achieved by the QAOA in the limit of infinite system size, as a function of the algorithm parameters. They also prove that typical measurements of random instances of the problem concentrates at this value. These results allow for comparisons to the state-of-the-art classical algorithms. In particular, the authors find that the QAOA with 11 layers outperforms the standard semidefinite programming algorithm on this problem. It remains an open question how the performance scaling of the QAOA compares to the currently known best classical algorithm by Montanari.

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► References

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