Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection

Authors Ryo Hiromasa, Akihiro Mizutani, Yuki Takeuchi, Seiichiro Tani

Thumbnail PDF


  • Filesize: 0.9 MB
  • 23 pages

Document Identifiers

Author Details

Ryo Hiromasa
  • Information Technology R&D Center, Mitsubishi Electric Corporation, Kamakura, Japan
Akihiro Mizutani
  • Information Technology R&D Center, Mitsubishi Electric Corporation, Kamakura, Japan
Yuki Takeuchi
  • NTT Communication Science Laboratories, NTT Corporation, Atsugi, Japan
Seiichiro Tani
  • NTT Communication Science Laboratories, NTT Corporation, Atsugi, Japan
  • International Research Frontiers Initiative (IRFI), Tokyo Institute of Technology, Japan


We thank Yasuhiro Takahashi and Yusuke Aikawa for helpful discussions. We also thank Tomoyuki Morimae for fruitful discussions and pointing out Refs. [Aaronson et al., 2014; Aaronson et al., 2016] to us.

Cite AsGet BibTex

Ryo Hiromasa, Akihiro Mizutani, Yuki Takeuchi, and Seiichiro Tani. Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We define rewinding operators that invert quantum measurements. Then, we define complexity classes RwBQP, CBQP, and AdPostBQP as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that BPP^PP ⊆ RwBQP = CBQP = AdPostBQP ⊆ PSPACE. As a byproduct of this result, we show that any problem in PostBQP can be solved with only postselections of outputs whose probabilities are polynomially close to one. Under the strongly believed assumption that BQP ⊉ SZK, or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. In addition, we consider rewindable Clifford and instantaneous quantum polynomial time circuits.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Quantum computing
  • Postselection
  • Lattice problems


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


  1. S. Aaronson. Quantum lower bound for the collision problem. In Proceedings of the 34th Annual Annual ACM Symposium on Theory of Computing, pages 635-642, 2002. Google Scholar
  2. S. Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2063):3473-3482, 2005. Google Scholar
  3. S. Aaronson and A. Arkhipov. The computational complexity of linear optics. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pages 333-342, 2011. Google Scholar
  4. S. Aaronson, A. Bouland, J. Fitzsimons, and M. Lee. The space "just above" bqp, 2014. arXiv:1412.6507. Google Scholar
  5. S. Aaronson, A. Bouland, J. Fitzsimons, and M. Lee. The space "just above" bqp. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pages 271-280, 2016. Google Scholar
  6. D. S. Abrams and S. Lloyd. Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #xxx VERBATIM x PLACEHOLDER xxx 408-1 xxx REDLOHECALP x MITABREV xxx problems. Phys. Rev. Lett., 81(18):3992, 1998. Google Scholar
  7. A. Ambainis, A. Rosmanis, and D. Unruh. Quantum attacks on classical proof systems: The hardness of quantum rewinding. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science, pages 474-483, 2014. Google Scholar
  8. S. Arora and B. Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. Google Scholar
  9. E. Bernstein and U. Vazirani. Quantum complexity theory. SIAM Journal on Computing, 26(5):1411-1473, 1997. Google Scholar
  10. S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven. Characterizing quantum supremacy in near-term devices. Nature Physics, 14(6):595-600, 2018. Google Scholar
  11. A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani. On the complexity and verification of quantum random circuit sampling. Nature Physics, 15(2):159-163, 2019. Google Scholar
  12. M. J. Bremner, R. Jozsa, and D. J. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467(2126):459-472, 2011. Google Scholar
  13. M. J. Bremner, A. Montanaro, and D. J. Shepherd. Average-case complexity versus approximate simulation of commuting quantum computations. Phys. Rev. Lett., 117(8):080501, 2016. Google Scholar
  14. A. Cojocaru, L. Colisson, E. Kashefi, and P. Wallden. Qfactory: Classically-instructed remote secret qubits preparation. In Proceedings of the 25th Annual International Conference on the Theory and Application of Cryptology and Information Security, pages 615-645, 2019. Google Scholar
  15. A. Cojocaru, L. Colisson, E. Kashefi, and P. Wallden. On the possibility of classical client blind quantum computing. Cryptography, 5(1):3, 2021. Google Scholar
  16. C. M. Dawson and M. A. Nielsen. The solovay-kitaev algorithm. Quantum Information and Computation, 6(1):81-95, 2006. Google Scholar
  17. K. Fujii, H. Kobayashi, T. Morimae, H. Nishimura, S. Tamate, and S. Tani. Impossibility of classically simulating one-clean-qubit model with multiplicative error. Phys. Rev. Lett., 120(20):200502, 2018. Google Scholar
  18. X. Gao, S.-T. Wang, and L.-M. Duan. Quantum supremacy for simulating a translation-invariant ising spin model. Phys. Rev. Lett., 118(4):040502, 2017. Google Scholar
  19. D. Gottesman. The heisenberg representation of quantum computers. In Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, pages 32-43, 1999. Google Scholar
  20. C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex. Gaussian boson sampling. Phys. Rev. Lett., 119(17):170501, 2017. Google Scholar
  21. D. Hangleiter, J. Bermejo-Vega, M. Schwarz, and J. Eisert. Anticoncentration theorems for schemes showing a quantum speedup. Quantum, 2:65, 2018. Google Scholar
  22. A. W. Harrow and A. Montanaro. Quantum computational supremacy. Nature, 549(7671):203-209, 2017. Google Scholar
  23. R. Hiromasa, A. Mizutani, Y. Takeuchi, and S. Tani. Rewindable quantum computation and its equivalence to cloning and adaptive postselection, 2022. arXiv:2206.05434. Google Scholar
  24. A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O'Brien, and T. C. Ralph. Boson sampling from a gaussian state. Phys. Rev. Lett., 113(10):100502, 2014. Google Scholar
  25. J. Miller, S. Sanders, and A. Miyake. Quantum supremacy in constant-time measurement-based computation: A unified architecture for sampling and verification. Phys. Rev. A, 96(6):062320, 2017. Google Scholar
  26. T. Morimae, K. Fujii, and J. F. Fitzsimons. Hardness of classically simulating the one-clean-qubit model. Phys. Rev. Lett., 112(13):130502, 2014. Google Scholar
  27. T. Morimae, Y. Takeuchi, and H. Nishimura. Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy. Quantum, 2:106, 2018. Google Scholar
  28. T. Morimae, Y. Takeuchi, and S. Tani. Sampling of globally depolarized random quantum circuit, 2019. arXiv:1911.02220. Google Scholar
  29. C. Peikert and V. Vaikuntanathan. Noninteractive statistical zero-knowledge proofs for lattice problems. In Proceedings of the 28th International Cryptology Conference, pages 536-553, 2008. Google Scholar
  30. O. Regev. On lattices, learning with errors, random linear codes, and cryptography. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 84-93, 2005. Google Scholar
  31. A. Sahai and S. Vadhan. A complete problem for statistical zero knowledge. Journal of the ACM, 50(2):196-249, 2003. Google Scholar
  32. Y. Shi. Both toffoli and controlled-not need little help to do universal quantum computation, 2002. arXiv:quant-ph/0205115. Google Scholar
  33. P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484-1509, 1997. Google Scholar
  34. Y. Takahashi, S. Tani, T. Yamazaki, and K. Tanaka. Commuting quantum circuits with few outputs are unlikely to be classically simulatable. Quantum Information and Computation, 16(3&4):251-270, 2016. Google Scholar
  35. Y. Takeuchi and Y. Takahashi. Ancilla-driven instantaneous quantum polynomial time circuit for quantum supremacy. Phys. Rev. A, 94(6):062336, 2016. Google Scholar
  36. D. Unruh. Computationally binding quantum commitments. In Proceedings of the 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 497-527, 2016. Google Scholar
  37. J. Watrous. Zero-knowledge against quantum attacks. SIAM Journal on Computing, 39(1):25-58, 2009. Google Scholar
  38. W. K. Wootters and W. H. Zurek. A single quantum cannot be cloned. Nature, 299(5886):802-803, 1982. Google Scholar