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

## Abstract

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.

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## 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 https://github.com/quantumlib/ReCirq.

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## Acknowledgements

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

Authors

### Contributions

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.

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

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). https://doi.org/10.1038/s41567-020-01105-y

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