On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation

Authors Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, Yu-Ching Shen

Thumbnail PDF


  • Filesize: 1.09 MB
  • 45 pages

Document Identifiers

Author Details

Nai-Hui Chia
  • Rice University, Houston, TX, USA
Kai-Min Chung
  • Academia Sinica, Taipei, Taiwan
Yao-Ching Hsieh
  • University of Washington, Seattle, WA, USA
Han-Hsuan Lin
  • National Tsing Hua University, Hsinchu, Taiwan
Yao-Ting Lin
  • University of California at Santa Barbara, CA, USA
Yu-Ching Shen
  • Academia Sinica, Taipei, Taiwan

Cite AsGet BibTex

Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, and Yu-Ching Shen. On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 33:1-33:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Hamiltonian simulation is one of the most important problems in the field of quantum computing. There have been extended efforts on designing algorithms for faster simulation, and the evolution time T for the simulation greatly affect algorithm runtime as expected. While there are some specific types of Hamiltonians that can be fast-forwarded, i.e., simulated within time o(T), for some large classes of Hamiltonians (e.g., all local/sparse Hamiltonians), existing simulation algorithms require running time at least linear in the evolution time T. On the other hand, while there exist lower bounds of Ω(T) circuit size for some large classes of Hamiltonian, these lower bounds do not rule out the possibilities of Hamiltonian simulation with large but "low-depth" circuits by running things in parallel. As a result, physical systems with system size scaling with T can potentially do a fast-forwarding simulation. Therefore, it is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism. In this work, we give a negative result for the above open problem in various settings. In the oracle model, we prove that there are time-independent sparse Hamiltonians that cannot be simulated via an oracle circuit of depth o(T). In the plain model, relying on the random oracle heuristic, we show that there exist time-independent local Hamiltonians and time-dependent geometrically local Hamiltonians on n qubits that cannot be simulated via an oracle circuit of depth o(T/n^c), where the Hamiltonians act on n qubits, and c is a constant. Lastly, we generalize the above results and show that any simulators that are geometrically local Hamiltonians cannot do the simulation much faster than parallel quantum algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Hamiltonian simulation
  • Depth lower bound
  • Parallel query lower bound


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


  1. Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev. Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation. SIAM Review, 50(4):755-787, 2008. Google Scholar
  2. Brian Alspach. Johnson graphs are Hamilton-connected. Ars Mathematica Contemporanea, 6(1):21-23, 2012. Google Scholar
  3. Yosi Atia and Dorit Aharonov. Fast-forwarding of hamiltonians and exponentially precise measurements. Nature Communications, 8(1), November 2017. URL: https://doi.org/10.1038/s41467-017-01637-7.
  4. Mihir Bellare and Phillip Rogaway. Random oracles are practical: A paradigm for designing efficient protocols. In Proceedings of the 1st ACM Conference on Computer and Communications Security, pages 62-73, 1993. Google Scholar
  5. Charles H Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and weaknesses of quantum computing. SIAM journal on Computing, 26(5):1510-1523, 1997. Google Scholar
  6. Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. Efficient quantum algorithms for simulating sparse Hamiltonians. Communications in Mathematical Physics, 270(2):359-371, 2007. Google Scholar
  7. Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Exponential improvement in precision for simulating sparse Hamiltonians. Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 283-292, May 2014. arXiv: 1312.1414. URL: https://doi.org/10.1145/2591796.2591854.
  8. Dominic W. Berry, Andrew M. Childs, and Robin Kothari. Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 792-809, October 2015. ISSN: 0272-5428. URL: https://doi.org/10.1109/FOCS.2015.54.
  9. Dan Boneh, Joseph Bonneau, Benedikt Bünz, and Ben Fisch. Verifiable delay functions. In CRYPTO (1), volume 10991 of Lecture Notes in Computer Science, pages 757-788. Springer, 2018. Google Scholar
  10. Dan Boneh, Özgür Dagdelen, Marc Fischlin, Anja Lehmann, Christian Schaffner, and Mark Zhandry. Random oracles in a quantum world. In Advances in Cryptology-ASIACRYPT 2011: 17th International Conference on the Theory and Application of Cryptology and Information Security, Seoul, South Korea, December 4-8, 2011. Proceedings 17, pages 41-69. Springer, 2011. Google Scholar
  11. P Oscar Boykin, Tal Mor, Matthew Pulver, Vwani Roychowdhury, and Farrokh Vatan. On universal and fault-tolerant quantum computing: a novel basis and a new constructive proof of universality for Shor’s basis. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 486-494. IEEE, 1999. Google Scholar
  12. Jorge Chávez-Saab, Francisco Rodríguez-Henríquez, and Mehdi Tibouchi. Verifiable isogeny walks: Towards an isogeny-based postquantum VDF. In SAC, volume 13203 of Lecture Notes in Computer Science, pages 441-460. Springer, 2021. Google Scholar
  13. Andrew M. Childs and Dominic W. Berry. Black-box Hamiltonian simulation and unitary implementation. Quantum Information and Computation, 12(1&2):29-62, January 2012. URL: https://doi.org/10.26421/QIC12.1-2-4.
  14. Andrew M. Childs, Richard Cleve, Enrico Deotto, Edward Farhi, Sam Gutmann, and Daniel A. Spielman. Exponential algorithmic speedup by a quantum walk. In Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC quotesingle03. ACM Press, 2003. URL: https://doi.org/10.1145/780542.780552.
  15. Andrew MacGregor Childs. Quantum Information Processing in Continuous Time. Thesis, Massachusetts Institute of Technology, 2004. Google Scholar
  16. Kai-Min Chung, Serge Fehr, Yu-Hsuan Huang, and Tai-Ning Liao. On the compressed-oracle technique, and post-quantum security of proofs of sequential work. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 598-629. Springer, 2021. Google Scholar
  17. J. Ignacio Cirac and Peter Zoller. Goals and opportunities in quantum simulation. Nature Physics, 8(4):264-266, April 2012. URL: https://doi.org/10.1038/nphys2275.
  18. Toby Cubitt, Ashley Montanaro, and Stephen Piddock. Universal Quantum Hamiltonians. Proceedings of the National Academy of Sciences, 115(38):9497-9502, September 2018. URL: https://doi.org/10.1073/pnas.1804949115.
  19. Naomi Ephraim, Cody Freitag, Ilan Komargodski, and Rafael Pass. Continuous verifiable delay functions. In EUROCRYPT (3), volume 12107 of Lecture Notes in Computer Science, pages 125-154. Springer, 2020. Google Scholar
  20. Luca De Feo, Simon Masson, Christophe Petit, and Antonio Sanso. Verifiable delay functions from supersingular isogenies and pairings. In ASIACRYPT (1), volume 11921 of Lecture Notes in Computer Science, pages 248-277. Springer, 2019. Google Scholar
  21. Richard P. Feynman. Quantum Mechanical Computers. Optics News, 11(2):11-20, February 1985. URL: https://doi.org/10.1364/ON.11.2.000011.
  22. Shouzhen Gu, Rolando D. Somma, and Burak Ş ahinoğlu. Fast-forwarding quantum evolution. Quantum, 5:577, November 2021. URL: https://doi.org/10.22331/q-2021-11-15-577.
  23. Jeongwan Haah, Matthew B. Hastings, Robin Kothari, and Guang Hao Low. Quantum algorithm for simulating real time evolution of lattice Hamiltonians. SIAM Journal on Computing, pages FOCS18-250-FOCS18-284, January 2021. URL: https://doi.org/10.1137/18m1231511.
  24. Stacey Jeffery, Frederic Magniez, and Ronald de Wolf. Optimal parallel quantum query algorithms, 2013. URL: https://doi.org/10.48550/ARXIV.1309.6116.
  25. A. Kitaev, A. Shen, and M. Vyalyi. Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, May 2002. URL: https://doi.org/10.1090/gsm/047.
  26. Ilia Krasikov. Uniform bounds for Bessel functions. Journal of Applied Analysis, 12(1):83-91, 2006. Google Scholar
  27. Guang Hao Low and Isaac L. Chuang. Optimal Hamiltonian simulation by quantum signal processing. Phys. Rev. Lett., 118:010501, January 2017. URL: https://doi.org/10.1103/PhysRevLett.118.010501.
  28. Guang Hao Low and Isaac L. Chuang. Hamiltonian Simulation by Qubitization. Quantum, 3:163, July 2019. URL: https://doi.org/10.22331/q-2019-07-12-163.
  29. Michael Luby and Charles Rackoff. How to construct pseudorandom permutations from pseudorandom functions. SIAM Journal on Computing, 17(2):373-386, 1988. Google Scholar
  30. Daniel Nagaj. Fast universal quantum computation with railroad-switch local Hamiltonians. Journal of Mathematical Physics, 51(6):062201, 2010. Google Scholar
  31. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. URL: https://doi.org/10.1017/CBO9780511976667.
  32. Krzysztof Pietrzak. Simple verifiable delay functions. In ITCS, volume 124 of LIPIcs, pages 60:1-60:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  33. Ronald L Rivest, Adi Shamir, and David A Wagner. Time-lock puzzles and timed-release crypto. Massachusetts Institute of Technology. Laboratory for Computer Science, 1996. Google Scholar
  34. J. J. Sakurai and Jim Napolitano. Modern Quantum Mechanics. Cambridge University Press, third edition, September 2020. URL: https://doi.org/10.1017/9781108587280.
  35. Benjamin Wesolowski. Efficient verifiable delay functions. J. Cryptol., 33(4):2113-2147, 2020. Google Scholar
  36. Christof Zalka. Grover’s quantum searching algorithm is optimal. Physical Review A, 60(4):2746, 1999. Google Scholar
  37. Mark Zhandry. A note on the quantum collision and set equality problems, 2013. URL: https://doi.org/10.48550/arXiv.1312.1027.
  38. Mark Zhandry. How to record quantum queries, and applications to quantum indifferentiability. In Annual International Cryptology Conference, pages 239-268. Springer, 2019. Google Scholar
  39. Zhicheng Zhang, Qisheng Wang, and Mingsheng Ying. Parallel quantum algorithm for Hamiltonian simulation. arXiv preprint, 2021. URL: https://arxiv.org/abs/2105.11889.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail