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.

Restrictions on realizable unitary operations imposed by symmetry and locality


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.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: LSQCs.
Fig. 2: Schematic relation between group of all symmetric unitaries (torus) and subgroup generated by LSQCs (blue curve).
Fig. 3: Scheme for local symmetric process tomography and measurement of l-body phases.
Fig. 4: Circumventing the no-go theorem with ancillary qubits.

Data availability

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


  1. Noether, E. Nachrichten der koniglichen gesellschaft der wissenschaften, gottingen, mathematisch-physikalische klasse 2. Invariante Variationsprobleme 235–257 (1918).

  2. Noether, E. Invariant variation problems. Transp. Theory Stat. Phys. 1, 186 (1971).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Lieb, E. H. & Robinson D. W. in Statistical Mechanics (Springer, 1972).

  4. DiVincenzo, D. P. Two-bit gates are universal for quantum computation. Phys. Rev. A 51, 1015 (1995).

    Article  ADS  Google Scholar 

  5. Lloyd, S. Almost any quantum logic gate is universal. Phys. Rev. Lett. 75, 346 (1995).

    Article  ADS  Google Scholar 

  6. Deutsch, D. E., Barenco, A. & Ekert, A. Universality in quantum computation. Proc. R. Soc. London A 449, 669 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Khemani, V., Vishwanath, A. & Huse, D. A. Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws. Phys. Rev. X 8, 031057 (2018).

    Google Scholar 

  8. Horodecki, M. & Oppenheim, J. Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun. 4, 1 (2013).

    Article  Google Scholar 

  9. Brandao, F. G., Horodecki, M., Oppenheim, J., Renes, J. M. & Spekkens, R. W. Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett. 111, 250404 (2013).

    Article  ADS  Google Scholar 

  10. Janzing, D., Wocjan, P., Zeier, R., Geiss, R. & Beth, T. Thermodynamic cost of reliability and low temperatures: tightening Landauer’s principle and the Second Law. Int. J. Theor. Phys. 39, 2717 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  11. Lostaglio, M., Korzekwa, K., Jennings, D. & Rudolph, T. Quantum coherence, time-translation symmetry, and thermodynamics. Phys. Rev. X 5, 021001 (2015).

    Google Scholar 

  12. Halpern, N. Y., Faist, P., Oppenheim, J. & Winter, A. Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges. Nat. Commun. 7, 12051 (2016).

    Article  ADS  Google Scholar 

  13. Halpern, N. Y. & Renes, J. M. Beyond heat baths: generalized resource theories for small-scale thermodynamics. Phys. Rev. E 93, 022126 (2016).

    Article  ADS  Google Scholar 

  14. Guryanova, Y., Popescu, S., Short, A. J., Silva, R. & Skrzypczyk, P. Thermodynamics of quantum systems with multiple conserved quantities. Nat. Commun. 7, ncomms12049 (2016).

    Article  ADS  Google Scholar 

  15. Chitambar, E. & Gour, G. Quantum resource theories. Rev. Mod. Phys. 91, 025001 (2019).

  16. Bartlett, S. D., Rudolph, T. & Spekkens, R. W. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys. 79, 555 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. d’Alessandro, D. Introduction to Quantum Control and Dynamics (CRC, 2007).

  18. Jurdjevic, V. & Sussmann, H. J. Control systems on Lie groups. J. Differ. Equ. 12, 313 (1972).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Brylinski J.-L. & Brylinski, R. in Universal Quantum Gates (Chapman and Hall, 2002).

  20. Childs, A. M., Leung, D., Mančinska, L., & Ozols, M. Characterization of universal two-qubit hamiltonians. Preprint at (2010).

  21. Zanardi, P. & Lloyd, S. Universal control of quantum subspaces and subsystems. Phys. Rev. A 69, 022313 (2004).

    Article  ADS  Google Scholar 

  22. Giorda, P., Zanardi, P. & Lloyd, S. Universal quantum control in irreducible state-space sectors: application to bosonic and spin-boson systems. Phys. Rev. A 68, 062320 (2003).

    Article  ADS  Google Scholar 

  23. Bacon, D., Kempe, J., Lidar, D. A. & Whaley, K. B. Universal fault-tolerant quantum computation on decoherence-free subspaces. Phys. Rev. Lett. 85, 1758 (2000).

    Article  ADS  Google Scholar 

  24. Lidar, D. A., Chuang, I. L. & Whaley, K. B. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett. 81, 2594 (1998).

    Article  ADS  Google Scholar 

  25. Nielsen M. A., & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2010).

  26. Jordan, P. & Wigner, E. P. About the Pauli exclusion principle. Z. Phys. 47, 14 (1928).

    Google Scholar 

  27. Fradkin, E. Jordan–Wigner transformation for quantum-spin systems in two dimensions and fractional statistics. Phys. Rev. Lett. 63, 322 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  28. Nielsen, M. A. et al. The Fermionic Canonical Commutation Relations and the Jordan–Wigner Transform (University of Queensland, 2005).

  29. Jonathan, D. & Plenio, M. B. Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett. 83, 3566 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Ball, H., Oliver, W. D. & Biercuk, M. J. The role of master clock stability in quantum information processing. npj Quantum Inf. 2, 1 (2016).

    Article  Google Scholar 

  31. Bermudez, A. et al. Assessing the progress of trapped-ion processors towards fault-tolerant quantum computation. Phys. Rev. X 7, 041061 (2017).

    Google Scholar 

  32. Zanardi, P. & Rasetti, M. Noiseless quantum codes. Phys. Rev. Lett. 79, 3306 (1997).

    Article  ADS  Google Scholar 

  33. Molmer, K. & Sorensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835 (1999).

    Article  ADS  Google Scholar 

  34. Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum principal component analysis. Nat. Phys. 10, 631 (2014).

    Article  Google Scholar 

  35. Marvian, I. & Lloyd, S. Universal quantum emulator. Preprint at (2016).

  36. Kimmel, S., Y.-Y. Lin, C., Low, G. H., Ozols, M. & Yoder, T. J. Hamiltonian simulation with optimal sample complexity. npj Quantum Inf. 3, 1 (2017).

    Article  Google Scholar 

  37. Pichler, H., Zhu, G., Seif, A., Zoller, P. & Hafezi, M. Measurement protocol for the entanglement spectrum of cold atoms. Phys. Rev. X 6, 041033 (2016).

    Google Scholar 

  38. Popescu, S., Sainz, A. B., Short, A. J. & Winter, A. Quantum reference frames and their applications to thermodynamics. Phil. Trans. R. Soc. A 376, 20180111 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Marvian, I. & Mann, R. Building all time evolutions with rotationally invariant hamiltonians. Phys. Rev. A 78, 022304 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  40. Faist, P. et al. Continuous symmetries and approximate quantum error correction. Phys. Rev. X 10, 041018 (2020).

    Google Scholar 

  41. Hayden, P., Nezami, S., Popescu, S. & Salton, G. Error correction of quantum reference frame information. PRX Quantum. 2, 010326 (2021).

    Article  Google Scholar 

  42. Kong, L. & Liu Z.-W. Charge-conserving unitaries typically generate optimal covariant quantum error-correcting codes. Preprint at (2021).

  43. Aaronson, S. The complexity of quantum states and transformations: from quantum money to black holes. Preprint at (2016).

  44. Chen, X., Gu, Z.-C. & Wen, X.-G. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B 82, 155138 (2010).

    Article  ADS  Google Scholar 

  45. Chen, X., Gu, Z.-C. & Wen, X.-G. Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B 83, 035107 (2011).

    Article  ADS  Google Scholar 

  46. Susskind, L. Computational complexity and black hole horizons. Fortschr. Phys. 64, 24 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  47. Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B. & Zhao, Y. Holographic complexity equals bulk action? Phys. Rev. Lett. 116, 191301 (2016).

    Article  ADS  Google Scholar 

  48. Stanford, D. & Susskind, L. Complexity and shock wave geometries. Phys. Rev. D 90, 126007 (2014).

    Article  ADS  Google Scholar 

  49. Banuls, M. C. et al. Simulating lattice gauge theories within quantum technologies. Eur. Phys. J. D 74, 1 (2020).

    Article  Google Scholar 

  50. Altman, E. et al. Quantum simulators: architectures and opportunities. PRX Quantum 2, 017003 (2021).

    Article  Google Scholar 

  51. Yang, B. et al. Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator. Nature 587, 392 (2020).

    Article  ADS  Google Scholar 

Download references


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 and Affiliations



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

Additional information

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

Supplementary information

Supplementary Information

Supplementary Notes 1–7

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Marvian, I. Restrictions on realizable unitary operations imposed by symmetry and locality. Nat. Phys. 18, 283–289 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


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