Parameter Setting in Quantum Approximate Optimization of Weighted Problems

Shree Hari Sureshbabu1, Dylan Herman1, Ruslan Shaydulin1, Joao Basso2, Shouvanik Chakrabarti1, Yue Sun1, and Marco Pistoia1

1Global Technology Applied Research, JPMorgan Chase, New York, NY 10017
2Department of Mathematics, University of California, Berkeley, CA 94720

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Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate algorithm for solving combinatorial optimization problems on quantum computers. However, in many cases QAOA requires computationally intensive parameter optimization. The challenge of parameter optimization is particularly acute in the case of weighted problems, for which the eigenvalues of the phase operator are non-integer and the QAOA energy landscape is not periodic. In this work, we develop parameter setting heuristics for QAOA applied to a general class of weighted problems. First, we derive optimal parameters for QAOA with depth $p=1$ applied to the weighted MaxCut problem under different assumptions on the weights. In particular, we rigorously prove the conventional wisdom that in the average case the first local optimum near zero gives globally-optimal QAOA parameters. Second, for $p\geq 1$ we prove that the QAOA energy landscape for weighted MaxCut approaches that for the unweighted case under a simple rescaling of parameters. Therefore, we can use parameters previously obtained for unweighted MaxCut for weighted problems. Finally, we prove that for $p=1$ the QAOA objective sharply concentrates around its expectation, which means that our parameter setting rules hold with high probability for a random weighted instance. We numerically validate this approach on general weighted graphs and show that on average the QAOA energy with the proposed fixed parameters is only $1.1$ percentage points away from that with optimized parameters. Third, we propose a general heuristic rescaling scheme inspired by the analytical results for weighted MaxCut and demonstrate its effectiveness using QAOA with the XY Hamming-weight-preserving mixer applied to the portfolio optimization problem. Our heuristic improves the convergence of local optimizers, reducing the number of iterations by 7.4x on average.

This work investigates parameter setting rules for QAOA, a leading quantum heuristic algorithm, applied to a general class of combinatorial optimization problems. Parameter optimization is a significant bottleneck towards near-term application. A general parameter-scaling heuristic for transferring QAOA parameters between weighted problem instances is proposed and rigorous results showing the effectiveness of this procedure on MaxCut is presented. Additionally, the numerics show that this procedure significantly reduces the training time of QAOA for portfolio optimization, which is an important problem in financial engineering

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Cited by

[1] Igor Gaidai and Rebekah Herrman, "Performance Analysis of Multi-Angle QAOA for p > 1", arXiv:2312.00200, (2023).

[2] Dylan Herman, Cody Googin, Xiaoyuan Liu, Yue Sun, Alexey Galda, Ilya Safro, Marco Pistoia, and Yuri Alexeev, "Quantum computing for finance", Nature Reviews Physics 5 8, 450 (2023).

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[4] Filip B. Maciejewski, Stuart Hadfield, Benjamin Hall, Mark Hodson, Maxime Dupont, Bram Evert, James Sud, M. Sohaib Alam, Zhihui Wang, Stephen Jeffrey, Bhuvanesh Sundar, P. Aaron Lott, Shon Grabbe, Eleanor G. Rieffel, Matthew J. Reagor, and Davide Venturelli, "Design and execution of quantum circuits using tens of superconducting qubits and thousands of gates for dense Ising optimization problems", arXiv:2308.12423, (2023).

[5] Dylan Herman, Ruslan Shaydulin, Yue Sun, Shouvanik Chakrabarti, Shaohan Hu, Pierre Minssen, Arthur Rattew, Romina Yalovetzky, and Marco Pistoia, "Constrained optimization via quantum Zeno dynamics", Communications Physics 6 1, 219 (2023).

[6] Ruslan Shaydulin, Changhao Li, Shouvanik Chakrabarti, Matthew DeCross, Dylan Herman, Niraj Kumar, Jeffrey Larson, Danylo Lykov, Pierre Minssen, Yue Sun, Yuri Alexeev, Joan M. Dreiling, John P. Gaebler, Thomas M. Gatterman, Justin A. Gerber, Kevin Gilmore, Dan Gresh, Nathan Hewitt, Chandler V. Horst, Shaohan Hu, Jacob Johansen, Mitchell Matheny, Tanner Mengle, Michael Mills, Steven A. Moses, Brian Neyenhuis, Peter Siegfried, Romina Yalovetzky, and Marco Pistoia, "Evidence of Scaling Advantage for the Quantum Approximate Optimization Algorithm on a Classically Intractable Problem", arXiv:2308.02342, (2023).

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