Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware

Johannes Weidenfeller1,2, Lucia C. Valor1, Julien Gacon1,3, Caroline Tornow1,2, Luciano Bello1, Stefan Woerner1, and Daniel J. Egger1

1IBM Quantum, IBM Research Europe – Zurich
2ETH Zurich
3Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL)

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Abstract

Quantum computers may provide good solutions to combinatorial optimization problems by leveraging the Quantum Approximate Optimization Algorithm (QAOA). The QAOA is often presented as an algorithm for noisy hardware. However, hardware constraints limit its applicability to problem instances that closely match the connectivity of the qubits. Furthermore, the QAOA must outpace classical solvers. Here, we investigate swap strategies to map dense problems into linear, grid and heavy-hex coupling maps. A line-based swap strategy works best for linear and two-dimensional grid coupling maps. Heavy-hex coupling maps require an adaptation of the line swap strategy. By contrast, three-dimensional grid coupling maps benefit from a different swap strategy. Using known entropic arguments we find that the required gate fidelity for dense problems lies deep below the fault-tolerant threshold. We also provide a methodology to reason about the execution-time of QAOA. Finally, we present a QAOA Qiskit Runtime program and execute the closed-loop optimization on cloud-based quantum computers with transpiler settings optimized for QAOA. This work highlights some obstacles to improve to make QAOA competitive, such as gate fidelity, gate speed, and the large number of shots needed. The Qiskit Runtime program gives us a tool to investigate such issues at scale on noisy superconducting qubit hardware.

Quantum computers have the potential to solve hard combinatorial optimization problems using quantum algorithms like the quantum approximate optimization algorithm (QAOA). Here, we investigate the hardware requirements to run QAOA on noisy quantum hardware. To reduce circuit depth, we introduce novel quantum gate routing strategies to map dense optimization problems to hardware with limited connectivity. We analyze the resulting requirements on the gate fidelity and present a framework to estimate the run-time of the QAOA. Finally, we demonstrate a Qiskit Runtime environment to execute quantum optimization on superconducting hardware with problem instances with 7 and 27 decision variables.

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Cited by

[1] Zheng-Hang Sun, Yong-Yi Wang, Jian Cui, and Heng Fan, "Improving the performance of quantum approximate optimization for preparing non-trivial quantum states without translational symmetry", New Journal of Physics 25 1, 013015 (2023).

[2] Pranav Chandarana, Narendra N. Hegade, Iraitz Montalban, Enrique Solano, and Xi Chen, "Digitized-Counterdiabatic Quantum Algorithm for Protein Folding", arXiv:2212.13511, (2022).

[3] Almudena Carrera Vazquez, Daniel J. Egger, David Ochsner, and Stefan Woerner, "Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation", arXiv:2207.11268, (2022).

[4] Maxime Dupont, Nicolas Didier, Mark J. Hodson, Joel E. Moore, and Matthew J. Reagor, "Entanglement perspective on the quantum approximate optimization algorithm", Physical Review A 106 2, 022423 (2022).

[5] Yanzhu Chen, Linghua Zhu, Chenxu Liu, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "How Much Entanglement Do Quantum Optimization Algorithms Require?", arXiv:2205.12283, (2022).

[6] Laurin E. Fischer, Daniel Miller, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Daniel J. Egger, and Ivano Tavernelli, "Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space", Physical Review Research 4 3, 033027 (2022).

[7] Maxime Dupont, Nicolas Didier, Mark J. Hodson, Joel E. Moore, and Matthew J. Reagor, "Calibrating the Classical Hardness of the Quantum Approximate Optimization Algorithm", PRX Quantum 3 4, 040339 (2022).

[8] Stefan H. Sack, Raimel A. Medina, Richard Kueng, and Maksym Serbyn, "Transition states and greedy exploration of the QAOA optimization landscape", arXiv:2209.01159, (2022).

[9] Nishant Jain, Brian Coyle, Elham Kashefi, and Niraj Kumar, "Graph neural network initialisation of quantum approximate optimisation", Quantum 6, 861 (2022).

[10] Cenk Tüysüz, Giuseppe Clemente, Arianna Crippa, Tobias Hartung, Stefan Kühn, and Karl Jansen, "Classical Splitting of Parametrized Quantum Circuits", arXiv:2206.09641, (2022).

[11] Gopal Chandra Santra, Fred Jendrzejewski, Philipp Hauke, and Daniel J. Egger, "Squeezing and quantum approximate optimization", arXiv:2205.10383, (2022).

[12] Pranav Chandarana, Pablo S. Vieites, Narendra N. Hegade, Enrique Solano, Yue Ban, and Xi Chen, "Meta-Learning Digitized-Counterdiabatic Quantum Optimization", arXiv:2206.09966, (2022).

[13] André Melo, Nathan Earnest-Noble, and Francesco Tacchino, "Pulse-efficient quantum machine learning", arXiv:2211.01383, (2022).

[14] Yuwei Jin, Jason Luo, Lucent Fong, Yanhao Chen, Ari B. Hayes, Chi Zhang, Fei Hua, and Eddy Z. Zhang, "A Structured Method for Compilation of QAOA Circuits in Quantum Computing", arXiv:2112.06143, (2021).

[15] Phillip C. Lotshaw, Kevin D. Battles, Bryan Gard, Gilles Buchs, Travis S. Humble, and Creston D. Herold, "Modelling noise in global Molmer-Sorensen interactions applied to quantum approximate optimization", arXiv:2211.00133, (2022).

[16] Christos Aravanis, Georgios Korpas, and Jakub Marecek, "Transpiling Quantum Circuits using the Pentagon Equation", arXiv:2209.14356, (2022).

[17] Phillip C. Lotshaw, Hanjing Xu, Bilal Khalid, Gilles Buchs, Travis S. Humble, and Arnab Banerjee, "Simulations of frustrated Ising Hamiltonians using quantum approximate optimization", Philosophical Transactions of the Royal Society of London Series A 381 2241, 20210414 (2023).

[18] Daniil Rabinovich, Ernesto Campos, Soumik Adhikary, Ekaterina Pankovets, Dmitry Vinichenko, and Jacob Biamonte, "On the gate-error robustness of variational quantum algorithms", arXiv:2301.00048, (2022).

[19] Atsushi Matsuo, Shigeru Yamashita, and Daniel J. Egger, "A SAT approach to the initial mapping problem in SWAP gate insertion for commuting gates", arXiv:2212.05666, (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2023-02-07 09:12:43) and SAO/NASA ADS (last updated successfully 2023-02-07 09:12:44). The list may be incomplete as not all publishers provide suitable and complete citation data.