Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quantum approximate optimization of non-planar graph problems on a planar superconducting processor


Faster algorithms for combinatorial optimization could prove transformative for diverse areas such as logistics, finance and machine learning. Accordingly, the possibility of quantum enhanced optimization has driven much interest in quantum technologies. Here we demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the planar connectivity graph native to our hardware; however, we also apply the QAOA to the Sherrington–Kirkpatrick model and MaxCut, non-native problems that require extensive compilation to implement. For hardware-native problems, which are classically efficient to solve on average, we obtain an approximation ratio that is independent of problem size and observe that performance increases with circuit depth. For problems requiring compilation, performance decreases with problem size. Circuits involving several thousand gates still present an advantage over random guessing but not over some efficient classical algorithms. Our results suggest that it will be challenging to scale near-term implementations of the QAOA for problems on non-native graphs. As these graphs are closer to real-world instances, we suggest more emphasis should be placed on such problems when using the QAOA to benchmark quantum processors.

This is a preview of subscription content

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Problem families under study.
Fig. 2: Circuits and compilation.
Fig. 3: Simulated and experimental QAOA landscapes.
Fig. 4: QAOA performance as a function of problem size n.
Fig. 5: QAOA performance as a function of depth p.

Data availability

Source data is available for this paper. The experimental data for this experiment is available from the Figshare repository39.

Code availability

The code used in this experiment is available38 with additional resources at


  1. 1.

    Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at (2014).

  2. 2.

    Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem. Preprint at (2014).

  3. 3.

    Biswas, R. et al. A NASA perspective on quantum computing: opportunities and challenges. Parallel Comput. 64, 81–98 (2017).

    MathSciNet  Article  Google Scholar 

  4. 4.

    Wecker, D., Hastings, M. B. & Troyer, M. Training a quantum optimizer. Phys. Rev. A 94, 022309 (2016).

    ADS  Article  Google Scholar 

  5. 5.

    Farhi, E. & Harrow, A. W. Quantum supremacy through the quantum approximate optimization algorithm. Preprint at (2016).

  6. 6.

    Jiang, Z., Rieffel, E. G. & Wang, Z. Near-optimal quantum circuit for Grover’s unstructured search using a transverse field. Phys. Rev. A 95, 062317 (2017).

    ADS  Article  Google Scholar 

  7. 7.

    Wang, Z., Hadfield, S., Jiang, Z. & Rieffel, E. G. Quantum approximate optimization algorithm for MaxCut: a fermionic view. Phys. Rev. A 97, 022304 (2018).

    ADS  Article  Google Scholar 

  8. 8.

    Hadfield, S. et al. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms 12, 34 (2017).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lloyd, S. Quantum approximate optimization is computationally universal. Preprint at (2018).

  10. 10.

    Farhi, E., Goldstone, J., Gutmann, S. & Zhou, L. The quantum approximate optimization algorithm and the Sherrington–Kirkpatrick model at infinite size. Preprint at (2019).

  11. 11.

    Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 5 (2014).

    Article  Google Scholar 

  12. 12.

    Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A 15, 3241–3253 (1982).

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Choi, V. Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7, 193–209 (2008).

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kadowaki, T. & Nishimori, H. Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355–5363 (1998).

    ADS  Article  Google Scholar 

  15. 15.

    Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001).

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Denchev, V. S. et al. What is the computational value of finite-range tunneling? Phys. Rev. X 6, 031015 (2016).

    Google Scholar 

  17. 17.

    Otterbach, J. S. et al. Unsupervised machine learning on a hybrid quantum computer. Preprint at (2017).

  18. 18.

    Willsch, M., Willsch, D., Jin, F., De Raedt, H. & Michielsen, K. Benchmarking the quantum approximate optimization algorithm. Quantum Inf. Process. 19, 197 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Abrams, D. M., Didier, N., Johnson, B. R., da Silva, M. P. & Ryan, C. A. Implementation of XY entangling gates with a single calibrated pulse. Nat. Electron. 3, 744–750 (2020).

    Article  Google Scholar 

  20. 20.

    Bengtsson, A. et al. Improved success probability with greater circuit depth for the quantum approximate optimization algorithm. Phys. Rev. Appl. 14, 034010 (2020).

    ADS  Article  Google Scholar 

  21. 21.

    Pagano, G. et al. Quantum approximate optimization of the long-range Ising model with a trapped-ion quantum simulator. Proc. Natl Acad. Sci. USA 117, 25396–25401 (2020).

    ADS  Article  Google Scholar 

  22. 22.

    Qiang, X. et al. Large-scale silicon quantum photonics implementing arbitrary two-qubit processing. Nat. Photon. 12, 534–539 (2018).

    ADS  Article  Google Scholar 

  23. 23.

    Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

    ADS  Article  Google Scholar 

  24. 24.

    Kelly, J., O’Malley, P., Neeley, M., Neven, H. & Martinis, J. M. Physical qubit calibration on a directed acyclic graph. Preprint at (2018).

  25. 25.

    Klimov, P. V., Kelly, J., Martinis, J. M. & Neven, H. The snake optimizer for learning quantum processor control parameters. Preprint at (2020).

  26. 26.

    The Cirq Developers Cirq: a python framework for creating, editing, and invoking noisy intermediate scale quantum (NISQ) circuits. GitHub (2020).

  27. 27.

    Rønnow, T. F. et al. Defining and detecting quantum speedup. Science 345, 420–424 (2014).

    ADS  Article  Google Scholar 

  28. 28.

    Goemans, M. X. & Williamson, D. P. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42, 1115–1145 (1995).

    MathSciNet  Article  Google Scholar 

  29. 29.

    Halperin, E., Livnat, D. & Zwick, U. Max Cut in cubic graphs. In Proc. Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’02 506–513 (Society for Industrial and Applied Mathematics, 2002).

  30. 30.

    Berman, P. & Karpinski, M. in Automata, Languages and Programming (eds Wiedermann, J. et al.) 200–209 (Springer, 1999).

  31. 31.

    Cowtan, A. et al. On the qubit routing problem. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019), Vol. 135 of Leibniz International Proc. Informatics (LIPIcs) (eds van Dam, W. & Mancinska, L.) 5:1–5:32 (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2019);

  32. 32.

    Sherrington, D. & Kirkpatrick, S. Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792–1796 (1975).

    ADS  Article  Google Scholar 

  33. 33.

    Montanari, A. Optimization of the Sherrington–Kirkpatrick Hamiltonian. Preprint at (2018).

  34. 34.

    Hirata, Y., Nakanishi, M., Yamashita, S. & Nakashima, Y. An efficient conversion of quantum circuits to a linear nearest neighbor architecture. Quantum Inf. Comput. 11, 142–166 (2011).

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Yang, Z.-C., Rahmani, A., Shabani, A., Neven, H. & Chamon, C. Optimizing variational quantum algorithms using Pontryagin’s minimum principle. Phys. Rev. X 7, 021027 (2017).

    Google Scholar 

  36. 36.

    McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).

    ADS  Article  Google Scholar 

  37. 37.

    Sung, K. J. et al. Using models to improve optimizers for variational quantum algorithms. Quantum Sci. Technol. (2020).

  38. 38.

    Google AI Quantum and collaborators Recirq. Zenodo (2020).

  39. 39.

    Google AI Quantum and collaborators. Sycamore QAOA experimental data. figshare (2020).

Download references


We thank the Cambridge Quantum Computing team for helpful correspondence about their \({\rm{t}}\left|{\rm{ket}}\right\rangle\) compiler, which we used for routing of MaxCut problems. The VW team acknowledges support from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 828826 ‘Quromorphic’. We thank all other members of the Google Quantum team, as well as our executive sponsors. D.B. is a CIFAR Associate Fellow in the Quantum Information Science Program.

Author information




R. Babbush and E.F. designed the experiment. M.P.H. and K.J.S. led code development and data collection with assistance from non-Google collaborators. Z.J. and N.C.R. derived the gate synthesis used in compilation. The manuscript was written by M.P.H., R. Babbush, E.F. and K.J.S. Experiments were performed using cloud access to a quantum processor that was recently developed and fabricated by a large effort involving the entire Google Quantum AI team.

Corresponding authors

Correspondence to Matthew P. Harrigan or Ryan Babbush.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Sections 1–6.

Source data

Source Data Fig. 4

Source Data for Fig. 4.

Source Data Fig. 5

Source Data for Fig. 5 (lines).

Source Data Fig. 5

Source Data for Fig. 5 (histogram).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Harrigan, M.P., Sung, K.J., Neeley, M. et al. Quantum approximate optimization of non-planar graph problems on a planar superconducting processor. Nat. Phys. 17, 332–336 (2021).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing