The quantum approximate optimization algorithm (QAOA) is a variational quantum algorithm, where a quantum computer implements a variational ansatz consisting of p layers of alternating unitary operators and a classical computer is used to optimize the variational parameters. For a random initialization, the optimization typically leads to local minima with poor performance, motivating the search for initialization strategies of QAOA variational parameters. Although numerous heuristic initializations exist, an analytical understanding and performance guarantees for large p remain evasive.We introduce a greedy initialization of QAOA which guarantees improving performance with an increasing number of layers. Our main result is an analytic construction of 2p + 1 transition states—saddle points with a unique negative curvature direction—for QAOA with p + 1 layers that use the local minimum of QAOA with p layers. Transition states connect to new local minima, which are guaranteed to lower the energy compared to the minimum found for p layers. We use the GREEDY procedure to navigate the exponentially increasing with p number of local minima resulting from the recursive application of our analytic construction. The performance of the GREEDY procedure matches available initialization strategies while providing a guarantee for the minimal energy to decrease with an increasing number of layers p.
Physical Review A
We thank V. Verteletskyi for a joint collaboration on numerical studies of the QAOA during his internship at ISTA that inspired analytic results on TS reported in this work. We acknowledge A. A. Mele and M. Brooks for discussions and D. Egger, P. Love, and D. Wierichs for valuable feedback on the manuscript. S.H.S., R.A.M., and M.S. acknowledge support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 850899). R.K. is supported by the SFB BeyondC (Grant No. F7107-N38) and the project QuantumReady (FFG 896217).
Sack S, Medina Ramos RA, Kueng R, Serbyn M. Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement. Physical Review A. 2023;107(6). doi:10.1103/physreva.107.062404
Sack, S., Medina Ramos, R. A., Kueng, R., & Serbyn, M. (2023). Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement. Physical Review A. American Physical Society. https://doi.org/10.1103/physreva.107.062404
Sack, Stefan, Raimel A Medina Ramos, Richard Kueng, and Maksym Serbyn. “Recursive Greedy Initialization of the Quantum Approximate Optimization Algorithm with Guaranteed Improvement.” Physical Review A. American Physical Society, 2023. https://doi.org/10.1103/physreva.107.062404.
S. Sack, R. A. Medina Ramos, R. Kueng, and M. Serbyn, “Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement,” Physical Review A, vol. 107, no. 6. American Physical Society, 2023.
Sack S, Medina Ramos RA, Kueng R, Serbyn M. 2023. Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement. Physical Review A. 107(6), 062404.
Sack, Stefan, et al. “Recursive Greedy Initialization of the Quantum Approximate Optimization Algorithm with Guaranteed Improvement.” Physical Review A, vol. 107, no. 6, 062404, American Physical Society, 2023, doi:10.1103/physreva.107.062404.
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