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# Classical algorithms for quantum mean values

## Abstract

Quantum algorithms hold the promise of solving certain computational problems dramatically faster than their classical counterparts. The latest generation of quantum processors with ~50 qubits are expected to be at the brink of outperforming classical computers. However, due to the lack of fault tolerance, the qubits can be operated for only a few time steps, making the quantum circuits shallow in depth. Variational quantum algorithms are leading candidates in the effort to find shallow-depth quantum algorithms that outperform classical computers. Here we consider the task of computing the mean values of multi-qubit observables, which is a cornerstone of variational quantum algorithms for optimization, machine learning and the simulation of quantum many-body systems. We develop sub-exponential time classical algorithms for solving the quantum mean value problem for general classes of quantum observables and constant-depth quantum circuits. In the special case of geometrically local two-dimensional quantum circuits, the runtime of our algorithm scales linearly with the number of qubits. Our results demonstrate that appropriate choices of circuit parameters such as geometric locality and depth are essential to obtain quantum speed-ups based on variational quantum algorithms.

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## Data availability

No datasets were generated or analysed during the current study.

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## Acknowledgements

We thank R. Koenig and K. Temme for helpful discussions. R.M. thanks A. Barvinok for discussions. D.G. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) under Discovery grant no. RGPIN-2019-04198. D.G. is a CIFAR fellow in the Quantum Information Science programme and is also supported in part by IBM Research. S.B. and R.M. acknowledge the support of the IBM Research Frontiers Institute and funding from the MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert Space.

## Author information

Authors

### Contributions

All authors contributed important ideas during initial discussions and contributed equally to deriving the technical proofs and writing the manuscript.

### Corresponding author

Correspondence to Ramis Movassagh.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks Xiao Yuan 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 text and table.

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Bravyi, S., Gosset, D. & Movassagh, R. Classical algorithms for quantum mean values. Nat. Phys. 17, 337–341 (2021). https://doi.org/10.1038/s41567-020-01109-8

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