Document Open Access Logo

Space-Bounded Unitary Quantum Computation with Postselection

Author Seiichiro Tani

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2022.81.pdf
  • Filesize: 0.75 MB
  • 15 pages

Document Identifiers

Author Details

Seiichiro Tani
  • NTT Communication Science Laboratories, NTT Corporation, Japan
  • International Research Frontiers Initiative (IRFI), Tokyo Institute of Technology, Japan

Funding

This work was partially supported by JSPS KAKENHI Grant Number JP20H05966 and JP22H00522.

Acknowledgements

I am grateful to anonymous referees of MFCS 2022 for their valuable comments.

Cite As Get BibTex

Seiichiro Tani. Space-Bounded Unitary Quantum Computation with Postselection. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 81:1-81:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.MFCS.2022.81

Abstract

Space-bounded computation has been a central topic in classical and quantum complexity theory. In the quantum case, every elementary gate must be unitary. This restriction makes it unclear whether the power of space-bounded computation changes by allowing intermediate measurement. In the bounded error case, Fefferman and Remscrim [STOC 2021, pp.1343-1356] and Girish, Raz and Zhan [ICALP 2021, pp.73:1-73:20] recently provided the break-through results that the power does not change. This paper shows that a similar result holds for space-bounded quantum computation with postselection. Namely, it is proved possible to eliminate intermediate postselections and measurements in the space-bounded quantum computation in the bounded-error setting. Our result strengthens the recent result by Le Gall, Nishimura and Yakaryilmaz [TQC 2021, pp.10:1-10:17] that logarithmic-space bounded-error quantum computation with intermediate postselections and measurements is equivalent in computational power to logarithmic-space unbounded-error probabilistic computation. As an application, it is shown that bounded-error space-bounded one-clean qubit computation (DQC1) with postselection is equivalent in computational power to unbounded-error space-bounded probabilistic computation, and the computational supremacy of the bounded-error space-bounded DQC1 is interpreted in complexity-theoretic terms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • quantum complexity theory
  • space-bounded computation
  • postselection

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Scott Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society A, 461(2063):3473-3482, 2005. URL: https://doi.org/10.1098/rspa.2005.1546.
  2. Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. Theory of Computing, 9:143-252, 2013. URL: https://doi.org/10.4086/toc.2013.v009a004.
  3. Andris Ambainis, Leonard J. Schulman, and Umesh V. Vazirani. Computing with highly mixed states. Journal of the ACM, 53(3):507-531, 2006. URL: https://doi.org/10.1145/1147954.1147962.
  4. R. Beigel, N. Reingold, and D. Spielman. PP is closed under intersection. Journal of Computer and System Sciences, 50(2):191-202, 1995. URL: https://doi.org/10.1006/jcss.1995.1017.
  5. A. Borodin, S. Cook, and N. Pippenger. Parallel computation for well-endowed rings and space-bounded probabilistic machines. Information and Control, 58(1):113-136, 1983. URL: https://doi.org/10.1016/S0019-9958(83)80060-6.
  6. Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proceedings of the Royal Society A, 467:459-472, 2010. URL: https://doi.org/10.1098/rspa.2010.0301.
  7. Bill Fefferman and Zachary Remscrim. Eliminating intermediate measurements in space-bounded quantum computation. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1343-1356, 2021. URL: https://doi.org/10.1145/3406325.3451051.
  8. Lance Fortnow and Nick Reingold. PP is closed under truth-table reductions. Inf. Comput., 124(1):1-6, 1996. URL: https://doi.org/10.1006/inco.1996.0001.
  9. Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Power of quantum computation with few clean qubits. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.13.
  10. Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Impossibility of classically simulating one-clean-qubit model with multiplicative error. Phys. Rev. Lett., 120:200502, 2018. URL: https://doi.org/10.1103/PhysRevLett.120.200502.
  11. François Le Gall, Harumichi Nishimura, and Abuzer Yakaryilmaz. Quantum logarithmic space and post-selection. In Proceedings of the 16th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2021, volume 197 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.TQC.2021.10.
  12. Uma Girish, Ran Raz, and Wei Zhan. Quantum logspace algorithm for powering matrices with bounded norm. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 73:1-73:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.73.
  13. Hermann Jung. On probabilistic time and space. In Proceedings of the 12th Colloquium on Automata, Languages and Programming, pages 310-317. Springer-Verlag, 1985. URL: https://doi.org/10.1007/BFb0015756.
  14. Alexei Yu. Kitaev, Alexander H. Shen, and Mikhail N. Vyalyi. Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. AMS, 2002. Google Scholar
  15. E. Knill and R. Laflamme. Power of one bit of quantum information. Phys. Rev. Lett., 81:5672-5675, December 1998. URL: https://doi.org/10.1103/PhysRevLett.81.5672.
  16. Dieter van Melkebeek and Thomas Watson. Time-space efficient simulations of quantum computations. Theory of Computing, 8(1):1-51, 2012. URL: https://doi.org/10.4086/toc.2012.v008a001.
  17. Tomoyuki Morimae, Keisuke Fujii, and Joseph F. Fitzsimons. Hardness of classically simulating the one-clean-qubit model. Phys. Rev. Lett., 112:130502, 2014. URL: https://doi.org/10.1103/PhysRevLett.112.130502.
  18. Tomoyuki Morimae, Keisuke Fujii, and Harumichi Nishimura. Power of one nonclean qubit. Phys. Rev. A, 95:042336, 2017. URL: https://doi.org/10.1103/PhysRevA.95.042336.
  19. Tomoyuki Morimae and Harumichi Nishimura. Merlinization of complexity classes above BQP. Quantum Info. Comput., 17(11-12):959-972, 2017. URL: https://doi.org/10.26421/QIC17.11-12-3.
  20. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. URL: https://doi.org/10.1017/CBO9780511976667.
  21. Harumichi Nishimura. Personal communication, 2021. Google Scholar
  22. M. Saks. Randomization and derandomization in space-bounded computation. In Proceedings of Computational Complexity (Formerly Structure in Complexity Theory), pages 128-149, 1996. URL: https://doi.org/10.1109/CCC.1996.507676.
  23. Peter W. Shor and Stephen P. Jordan. Estimating Jones polynomials is a complete problem for one clean qubit. Quantum Information & Computation, 8(8):681-714, 2008. URL: https://doi.org/10.26421/QIC8.8-9-1.
  24. Amnon Ta-Shma. Inverting well conditioned matrices in quantum logspace. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 881-890, 2013. URL: https://doi.org/10.1145/2488608.2488720.
  25. John Watrous. Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer and System Sciences, 62(2):376-391, 2001. URL: https://doi.org/10.1006/jcss.2000.1732.
  26. John Watrous. On the complexity of simulating space-bounded quantum computations. Computational Complexity, 12:48-84, 2003. URL: https://doi.org/10.1007/s00037-003-0177-8.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact