Document Open Access Logo

Unitary Complexity and the Uhlmann Transformation Problem

Authors John Bostanci , Yuval Efron , Tony Metger , Alexander Poremba , Luowen Qian , Henry Yuen

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2026.24.pdf
  • Filesize: 0.82 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Uhlmann’s theorem
  • unitary complexity theory

Metrics

Abstract

State transformation problems such as compressing quantum information or breaking quantum commitments are fundamental quantum tasks. However, their computational difficulty cannot easily be characterized using traditional complexity theory, which focuses on tasks with classical inputs and outputs. 
To study the complexity of such state transformation tasks, we introduce a framework for unitary synthesis problems, including notions of reductions and unitary complexity classes. We use this framework to study the complexity of transforming one entangled state into another via local operations. We formalize this as the Uhlmann Transformation Problem, an algorithmic version of Uhlmann’s theorem. Then, we prove structural results relating the complexity of the Uhlmann Transformation Problem, polynomial space quantum computation, and zero knowledge protocols. 
The Uhlmann Transformation Problem allows us to characterize the complexity of a variety of tasks in quantum information processing, including decoding noisy quantum channels, breaking falsifiable quantum cryptographic assumptions, implementing optimal prover strategies in quantum interactive proofs, and decoding the Hawking radiation of black holes. Our framework for unitary complexity thus provides new avenues for studying the computational complexity of many natural quantum information processing tasks.

Cite As Get BibTex

John Bostanci, Yuval Efron, Tony Metger, Alexander Poremba, Luowen Qian, and Henry Yuen. Unitary Complexity and the Uhlmann Transformation Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.24

Author Details

John Bostanci
  • Department of Computer Science, Columbia University, New York, NY, USA
Yuval Efron
  • Department of Computer Science, Columbia University, New York, NY, USA
Tony Metger
  • Institute for Theoretical Physics, ETH Zurich, Switzerland
Alexander Poremba
  • Department of Computer Science and Department of Physics, Boston University, MA, USA
Luowen Qian
  • CIS Lab, NTT Research, Sunnyvale, CA, USA
Henry Yuen
  • Department of Computer Science, Columbia University, New York, NY, USA

Funding

JB and HY are supported by AFOSR award FA9550-21-1-0040, NSF CAREER award CCF-2144219, and the Sloan Foundation. TM acknowledges support from SNSF Project Grant No. 200021_188541 and AFOSR-Grant No. FA9550-19-1-0202. AP is partially supported by AFOSR YIP (award number FA9550-16-1-0495), the Institute for Quantum Information and Matter (an NSF Physics Frontiers Center; NSF Grant PHY-1733907) and by a grant from the Simons Foundation (828076, TV). LQ is supported by DARPA under Agreement No. HR00112020023. We thank the Simons Institute for the Theory of Computing, where some of this work was conducted.

Acknowledgements

We thank Anurag Anshu, Lijie Chen, Andrea Coladangelo, Sam Gunn, Yunchao Liu, Joe Renes, Renato Renner, and Mehrdad Tahmasbi for helpful discussions. We thank Fred Dupuis for his help with understanding the decoupling results in his thesis. We are especially grateful to William Kretschmer, Fermi Ma, and John Wright for their thorough discussions on the first draft of the paper. We also thank Tomoyuki Morimae for his helpful comments on a preliminary version of this work. We thank anonymous conference reviewers for their helpful feedback.

References

  1. Scott Aaronson. The learnability of quantum states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2088):3089-3114, 2007. URL: https://doi.org/10.1098/rspa.2007.0113.
  2. Scott Aaronson. The complexity of quantum states and transformations: From quantum money to black holes, 2016. URL: https://arxiv.org/abs/1607.05256.
  3. Scott Aaronson. The Complexity Zoo. https://complexityzoo.net/Complexity_Zoo, 2023. Accessed: 2023-04-01.
  4. Scott Aaronson and Greg Kuperberg. Quantum versus classical proofs and advice. Theory of Computing, 3(7):129-157, 2007. URL: https://doi.org/10.4086/toc.2007.v003a007.
  5. Anura Abeyesinghe, Igor Devetak, Patrick Hayden, and Andreas Winter. The mother of all protocols: Restructuring quantum information’s family tree. Proceedings: Mathematical, Physical and Engineering Sciences, 465(2108):2537-2563, 2009. URL: https://doi.org/10.1098/rspa.2009.0202.
  6. Dorit Aharonov, Jordan Cotler, and Xiao-Liang Qi. Quantum algorithmic measurement. Nature Communications, 13(1):887, 2022. URL: https://doi.org/10.1038/s41467-021-27922-0.
  7. Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully. Black holes: complementarity or firewalls? Journal of High Energy Physics, 2013(2):62, February 2013. URL: https://doi.org/10.1007/JHEP02(2013)062.
  8. Prabhanjan Ananth, Luowen Qian, and Henry Yuen. Cryptography from pseudorandom quantum states. In Yevgeniy Dodis and Thomas Shrimpton, editors, Advances in Cryptology - CRYPTO 2022, pages 208-236, Cham, 2022. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-15802-5_8.
  9. Anurag Anshu and Srinivasan Arunachalam. A survey on the complexity of learning quantum states. Nature Reviews Physics, 6(1):59-69, January 2024. URL: https://doi.org/10.1038/s42254-023-00662-4.
  10. Anurag Anshu, Rahul Jain, and Naqueeb Ahmad Warsi. A one-shot achievability result for quantum state redistribution. IEEE Transactions on Information Theory, 64(3):1425-1435, 2018. URL: https://doi.org/10.1109/TIT.2017.2776112.
  11. Rotem Arnon-Friedman, Zvika Brakerski, and Thomas Vidick. Computational entanglement theory, 2023. URL: https://arxiv.org/abs/2310.02783.
  12. Mario Berta, Matthias Christandl, and Renato Renner. The quantum reverse shannon theorem based on one-shot information theory. Communications in Mathematical Physics, 306(3):579-615, 2011. URL: https://doi.org/10.1007/s00220-011-1309-7.
  13. John Bostanci, Yuval Efron, Tony Metger, Alexander Poremba, Luowen Qian, and Henry Yuen. Unitary complexity and the uhlmann transformation problem. arXiv preprint arXiv:2306.13073, 2023. URL: https://doi.org/10.48550/arXiv.2306.13073.
  14. Adam Bouland, Bill Fefferman, and Umesh Vazirani. Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality (Abstract). In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.63.
  15. Zvika Brakerski. Black-hole radiation decoding is quantum cryptography. In Helena Handschuh and Anna Lysyanskaya, editors, Advances in Cryptology - CRYPTO 2023, pages 37-65, Cham, 2023. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-38554-4_2.
  16. Zvika Brakerski, Ran Canetti, and Luowen Qian. On the Computational Hardness Needed for Quantum Cryptography. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.24.
  17. Adam R Brown, Daniel A Roberts, Leonard Susskind, Brian Swingle, and Ying Zhao. Complexity, action, and black holes. Physical Review D, 93:086006, April 2016. URL: https://doi.org/10.1103/PhysRevD.93.086006.
  18. Costin Bădescu and Ryan O'Donnell. Improved quantum data analysis. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1398-1411, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451109.
  19. Frédéric Dupuis. The decoupling approach to quantum information theory. PhD thesis, Université de Montréal, 2009. URL: https://doi.org/10.71781/10720.
  20. Craig Gentry and Daniel Wichs. Separating succinct non-interactive arguments from all falsifiable assumptions. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC '11, pages 99-108, New York, NY, USA, 2011. Association for Computing Machinery. URL: https://doi.org/10.1145/1993636.1993651.
  21. Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM Journal on Computing, 18(1):186-208, 1989. URL: https://doi.org/10.1137/0218012.
  22. Daniel Harlow and Patrick Hayden. Quantum computation vs. firewalls. Journal of High Energy Physics, 2013(6):85, 2013. URL: https://doi.org/10.1007/JHEP06(2013)085.
  23. Stephen W Hawking. Breakdown of predictability in gravitational collapse. Physical Review D, 14:2460-2473, 1976. URL: https://doi.org/10.1103/PhysRevD.14.2460.
  24. Patrick Hayden, Michał Horodecki, Andreas Winter, and Jon Yard. A decoupling approach to the quantum capacity. Open Systems & Information Dynamics, 15(01):7-19, 2008. URL: https://doi.org/10.1142/S1230161208000043.
  25. Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16(10):1050-1057, 2020. URL: https://doi.org/10.1038/s41567-020-0932-7.
  26. Russell Impagliazzo. A personal view of average-case complexity. In Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference, pages 134-147, 1995. URL: https://doi.org/10.1109/SCT.1995.514853.
  27. Russell Impagliazzo and Michael Luby. One-way functions are essential for complexity based cryptography. In 30th Annual Symposium on Foundations of Computer Science, pages 230-235, 1989. URL: https://doi.org/10.1109/SFCS.1989.63483.
  28. Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, and John Watrous. QIP = PSPACE. Journal of the ACM, 58(6), December 2011. URL: https://doi.org/10.1145/2049697.2049704.
  29. Zhengfeng Ji, Yi-Kai Liu, and Fang Song. Pseudorandom quantum states. In Hovav Shacham and Alexandra Boldyreva, editors, Advances in Cryptology - CRYPTO 2018, pages 126-152, Cham, 2018. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-96878-0_5.
  30. Elham Kashefi and Carolina Moura Alves. On the complexity of quantum languages, 2004. URL: https://arxiv.org/abs/quant-ph/0404062.
  31. Sumeet Khatri and Mark M. Wilde. Principles of quantum communication theory: A modern approach, 2020. URL: https://arxiv.org/abs/2011.04672.
  32. Dakshita Khurana and Kabir Tomer. Commitments from quantum one-wayness, 2023. URL: https://doi.org/10.48550/arXiv.2310.11526.
  33. Alexei Kitaev and John Watrous. Parallelization, amplification, and exponential time simulation of quantum interactive proof systems. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC '00, pages 608-617, New York, NY, USA, 2000. Association for Computing Machinery. URL: https://doi.org/10.1145/335305.335387.
  34. William Kretschmer. Quantum Pseudorandomness and Classical Complexity. In Min-Hsiu Hsieh, editor, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021), volume 197 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2021.2.
  35. William Kretschmer, Luowen Qian, Makrand Sinha, and Avishay Tal. Quantum cryptography in algorithmica. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1589-1602, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585225.
  36. Yanyi Liu and Rafael Pass. On one-way functions and kolmogorov complexity. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 1243-1254, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00118.
  37. Hoi-Kwong Lo and H.F. Chau. Why quantum bit commitment and ideal quantum coin tossing are impossible. Physica D: Nonlinear Phenomena, 120(1):177-187, 1998. Proceedings of the Fourth Workshop on Physics and Consumption. URL: https://doi.org/10.1016/S0167-2789(98)00053-0.
  38. Alex Lombardi, Fermi Ma, and John Wright. A one-query lower bound for unitary synthesis and breaking quantum cryptography, 2023. URL: https://doi.org/10.48550/arXiv.2310.08870.
  39. Dominic Mayers. Unconditionally secure quantum bit commitment is impossible. Physical Review Letters, 78:3414-3417, 1997. URL: https://doi.org/10.1103/PhysRevLett.78.3414.
  40. Tony Metger and Henry Yuen. stateQIP = statePSPACE. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1349-1356, Los Alamitos, CA, USA, November 2023. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS57990.2023.00082.
  41. Tomoyuki Morimae and Takashi Yamakawa. Quantum commitments and signatures without one-way functions. In Yevgeniy Dodis and Thomas Shrimpton, editors, Advances in Cryptology - CRYPTO 2022, pages 269-295, Cham, 2022. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-15802-5_10.
  42. Moni Naor. Bit commitment using pseudorandomness. Journal of Cryptology, 4:151-158, 1991. URL: https://doi.org/10.1007/BF00196774.
  43. Moni Naor. On cryptographic assumptions and challenges. In Dan Boneh, editor, Advances in Cryptology - CRYPTO 2003, pages 96-109, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-45146-4_6.
  44. Rafail Ostrovsky. One-way functions, hard on average problems, and statistical zero-knowledge proofs. In SCT, pages 133-138. Citeseer, 1991. URL: https://doi.org/10.1109/SCT.1991.160253.
  45. Christos H. Papadimitriou. NP-completeness: A retrospective. In Pierpaolo Degano, Roberto Gorrieri, and Alberto Marchetti-Spaccamela, editors, Automata, Languages and Programming, pages 2-6, Berlin, Heidelberg, 1997. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/3-540-63165-8_160.
  46. John Preskill. Do black holes destroy information? In Proceedings of the International Symposium on Black Holes, Membranes, Wormholes and Superstrings, pages 22-39. World Scientific, 1992. URL: https://doi.org/10.1142/9789814536752.
  47. Joseph M. Renes. Quantum Information Theory. De Gruyter Oldenbourg, Berlin, Boston, 2022. URL: https://doi.org/10.1515/9783110570250.
  48. Gregory Rosenthal. Efficient quantum state synthesis with one query. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2508-2534. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.89.
  49. Gregory Rosenthal and Henry Yuen. Interactive Proofs for Synthesizing Quantum States and Unitaries. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), volume 215 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.112.
  50. Amit Sahai and Salil Vadhan. A complete problem for statistical zero knowledge. Journal of the ACM, 50(2):196-249, March 2003. URL: https://doi.org/10.1145/636865.636868.
  51. Benjamin Schumacher. Quantum coding. Physical Review A, 51:2738-2747, April 1995. URL: https://doi.org/10.1103/PhysRevA.51.2738.
  52. Claude E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27(3):379-423, 1948. URL: https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.
  53. Leonard Susskind. Computational complexity and black hole horizons. Fortschritte der Physik, 64(1):24-43, 2016. URL: https://doi.org/10.1002/prop.201500092.
  54. Marco Tomamichel. A framework for non-asymptotic quantum information theory, 2013. URL: https://arxiv.org/abs/1203.2142.
  55. Armin Uhlmann. The "transition probability" in the state space of a *-algebra. Reports on Mathematical Physics, 9(2):273-279, 1976. URL: https://doi.org/10.1016/0034-4877(76)90060-4.
  56. John Watrous. Limits on the power of quantum statistical zero-knowledge. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pages 459-468, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181970.
  57. John Watrous. Zero-knowledge against quantum attacks. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 296-305, New York, NY, USA, 2006. Association for Computing Machinery. URL: https://doi.org/10.1145/1132516.1132560.
  58. Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2 edition, 2017. URL: https://doi.org/10.1017/CBO9781139525343.
  59. Jun Yan. General properties of quantum bit commitments (extended abstract). In Shweta Agrawal and Dongdai Lin, editors, Advances in Cryptology - ASIACRYPT 2022, pages 628-657, Cham, 2022. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-22972-5_22.
  60. Lisa Yang and Netta Engelhardt. The complexity of learning (pseudo)random dynamics of black holes and other chaotic systems, 2023. URL: https://arxiv.org/abs/2302.11013.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact