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# Limitations of optimization algorithms on noisy quantum devices

## Abstract

Recent successes in producing intermediate-scale quantum devices have focused interest on establishing whether near-term devices could outperform classical computers for practical applications. A central question is whether noise can be overcome in the absence of quantum error correction or if it fundamentally restricts any potential quantum advantage. We present a transparent way of comparing classical and quantum algorithms running on noisy devices for a large family of tasks that includes optimization and variational eigenstate solving. Our approach is based on entropic inequalities that determine how fast the quantum state converges to the fixed point of the noise model, together with established classical methods of Gibbs state simulation. Our techniques are extremely versatile and so may be applied to a large variety of algorithms, noise models and quantum computing architectures. We use our result to provide estimates for problems within reach of current experiments, such as quantum annealers or variational quantum algorithms. The bounds we obtain indicate that substantial quantum advantages are unlikely for classical optimization unless noise rates are decreased by orders of magnitude or the topology of the problem matches that of the device. This is the case even if the number of available qubits increases substantially.

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## Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

## Code availability

The code for this study can be found at ref. 62.

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## Acknowledgements

D.S.F. was supported by VILLUM FONDEN via the QMATH Centre of Excellence under grant no. 10059. R.G.-P. was supported by the Quantum Computing and Simulation Hub, an EPSRC-funded project, part of the UK National Quantum Technologies Programme. We thank H. Guo, J. Brown-Cohen and P.-L. Dallaire-Demers for helpful discussions.

## Author information

Authors

### Contributions

Both authors discussed the results and contributed to the writing and theory development for the final manuscript.

### Corresponding authors

Correspondence to Daniel Stilck França or Raul García-Patrón.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review informationNature Physics thanks Marco Tomamichel and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary information

Supplementary Discussion and Fig. 1.

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Stilck França, D., García-Patrón, R. Limitations of optimization algorithms on noisy quantum devices. Nat. Phys. 17, 1221–1227 (2021). https://doi.org/10.1038/s41567-021-01356-3