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Pseudo-Deterministic Quantum Algorithms

Authors Hugo Aaronson , Tom Gur , Jiawei Li

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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Pseudo-determinism
  • Quantum Computing
  • Algorithms

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Abstract

We initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions include the following complexity separations, which require new lower bound techniques specifically tailored to pseudo-determinism:
- We exhibit a problem, Avoid One Encrypted String (AOES), whose classical randomized query complexity is O(1) but is maximally hard for pseudo-deterministic quantum algorithms (Ω(N) query complexity). 
- We exhibit a problem, Quantum-Locked Estimation (QL-Estimation), for which pseudo-deterministic quantum algorithms admit an exponential speed-up over classical pseudo-deterministic algorithms (O(log(N)) vs. Θ(√N)), while the randomized query complexity is O(1). 
Complementing these separations, we show that for any total problem R, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., 𝖣(R) = Õ(psQ(R)⁵). On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead, including Grover search, element distinctness, triangle finding, k-sum, and graph collision.

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Hugo Aaronson, Tom Gur, and Jiawei Li. Pseudo-Deterministic Quantum Algorithms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.3

Author Details

Hugo Aaronson
  • Computing Laboratory, University of Cambridge, UK
Tom Gur
  • Computing Laboratory, University of Cambridge, UK
Jiawei Li
  • University of Texas at Austin, TX, USA

Funding

This research was supported by ERC Starting Grant 101163189 and UKRI Future Leaders Fellowship MR/X023583/1.

Acknowledgements

We thank Rahul Santhanam, Igor Oliveira, and Scott Aaronson for useful discussions on pseudo-determinism.

References

  1. Scott Aaronson. BQP and the polynomial hierarchy. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 141-150. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806711.
  2. Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, and Avishay Tal. Degree vs. approximate degree and quantum implications of huang’s sensitivity theorem. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1330-1342. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451047.
  3. Scott Aaronson and Shih-Han Hung. Certified randomness from quantum supremacy. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 933-944. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585145.
  4. Roozbeh Bassirian, Adam Bouland, Bill Fefferman, Sam Gunn, and Avishay Tal. On certified randomness from fourier sampling or random circuit sampling. arXiv preprint, 2021. URL: https://arxiv.org/abs/2111.14846.
  5. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  6. Vladimir Braverman, Robert Krauthgamer, Aditya Krishnan, and Shay Sapir. Lower bounds for pseudo-deterministic counting in a stream. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 30:1-30:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.30.
  7. Arkadev Chattopadhyay, Yogesh Dahiya, and Meena Mahajan. Query complexity of search problems. Electron. Colloquium Comput. Complex., TR23-039, 2023. URL: https://eccc.weizmann.ac.il/report/2023/039.
  8. Lijie Chen, Zhenjian Lu, Igor C. Oliveira, Hanlin Ren, and Rahul Santhanam. Polynomial-time pseudodeterministic construction of primes. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1261-1270. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00074.
  9. Andrew M. Childs and Jason M. Eisenberg. Quantum algorithms for subset finding. Quantum Inf. Comput., 5(7):593-604, 2005. URL: https://doi.org/10.26421/QIC5.7-7.
  10. Peter Dixon, Aduri Pavan, and N. V. Vinodchandran. Promise problems meet pseudodeterminism. CoRR, abs/2103.08589, 2021. URL: https://arxiv.org/abs/2103.08589.
  11. Peter Dixon, Aduri Pavan, Jason Vander Woude, and N. V. Vinodchandran. Pseudodeterminism: promises and lowerbounds. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1552-1565. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520043.
  12. Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Limit on the speed of quantum computation in determining parity. Physical Review Letters, 81(24):5442, 1998. Google Scholar
  13. Eran Gat and Shafi Goldwasser. Probabilistic search algorithms with unique answers and their cryptographic applications. Electron. Colloquium Comput. Complex., TR11-136, 2011. URL: https://eccc.weizmann.ac.il/report/2011/136.
  14. Michel X. Goemans, Shafi Goldwasser, and Dhiraj Holden. Doubly-efficient pseudo-deterministic proofs. CoRR, abs/1910.00994, 2019. URL: https://arxiv.org/abs/1910.00994.
  15. Oded Goldreich, Shafi Goldwasser, and Dana Ron. On the possibilities and limitations of pseudodeterministic algorithms. In Robert D. Kleinberg, editor, Innovations in Theoretical Computer Science, ITCS '13, Berkeley, CA, USA, January 9-12, 2013, pages 127-138. ACM, 2013. URL: https://doi.org/10.1145/2422436.2422453.
  16. Shafi Goldwasser, Ofer Grossman, and Dhiraj Holden. Pseudo-deterministic proofs. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 17:1-17:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.17.
  17. Shafi Goldwasser, Ofer Grossman, Sidhanth Mohanty, and David P. Woodruff. Pseudo-deterministic streaming. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 79:1-79:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.79.
  18. Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, and Rahul Santhanam. On the pseudo-deterministic query complexity of NP search problems. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 36:1-36:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.36.
  19. Ofer Grossman, Meghal Gupta, and Mark Sellke. Tight space lower bound for pseudo-deterministic approximate counting. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1496-1504. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00091.
  20. Robin Kothari. An optimal quantum algorithm for the oracle identification problem. In Ernst W. Mayr and Natacha Portier, editors, 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014, Lyon, France, March 5-8, 2014, LIPIcs, pages 482-493. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2014. URL: https://doi.org/10.4230/LIPIcs.STACS.2014.482.
  21. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information (10th Anniversary edition). Cambridge University Press, 2016. URL: https://www.cambridge.org/de/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/quantum-computation-and-quantum-information-10th-anniversary-edition?format=HB.
  22. Daniel R. Simon. On the power of quantum computation. SIAM J. Comput., 26(5):1474-1483, 1997. URL: https://doi.org/10.1137/S0097539796298637.
  23. Takashi Yamakawa and Mark Zhandry. Verifiable quantum advantage without structure. J. ACM, 71(3):20, 2024. URL: https://doi.org/10.1145/3658665.
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