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# Restrictions on realizable unitary operations imposed by symmetry and locality

## Abstract

According to a fundamental result in quantum computing, any unitary transformation on a composite system can be generated using so-called 2-local unitaries that act only on two subsystems. Beyond its importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints on the short-term dynamics, it does not restrict the possible unitary evolutions that a composite system with a general local Hamiltonian can experience after a sufficiently long time. Here we show that this universality does not remain valid in the presence of conservation laws and global continuous symmetries such as U(1) and SU(2). In particular, we show that generic symmetric unitaries cannot be implemented, even approximately, using local symmetric unitaries. Based on this no-go theorem, we propose a method for experimentally probing the locality of interactions in nature. In the context of quantum thermodynamics, our results mean that generic energy-conserving unitary transformations on a composite system cannot be realized solely by combining local energy-conserving unitaries on the components. We show how this can be circumvented via catalysis.

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Data sharing is not applicable to this article, as no datasets were generated or analysed during the current study.

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## Acknowledgements

I thank A. Hulse, D. Jennings, H. Liu, H. Salmasian and N. Yunger-Halpern for reading the manuscript carefully and providing many useful comments. This work was supported by NSF FET-1910571, NSF Phy-2046195 and Army Research Office (W911NF-21-1-0005).

## Author information

Authors

### Contributions

I.M. was the sole contributor to all aspects of this work.

### Corresponding author

Correspondence to Iman Marvian.

## Ethics declarations

### Competing interests

The author declares no competing interest.

## Peer review information

Nature Physics thanks Álvaro Alhambra 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 Notes 1–7

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Marvian, I. Restrictions on realizable unitary operations imposed by symmetry and locality. Nat. Phys. 18, 283–289 (2022). https://doi.org/10.1038/s41567-021-01464-0

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