Document Open Access Logo

The Parametrized Complexity of Quantum Verification

Authors Srinivasan Arunachalam, Sergey Bravyi , Chinmay Nirkhe , Bryan O'Gorman



PDF
Thumbnail PDF

File

LIPIcs.TQC.2022.3.pdf
  • Filesize: 0.79 MB
  • 18 pages

Document Identifiers

Author Details

Srinivasan Arunachalam
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA
Sergey Bravyi
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA
Chinmay Nirkhe
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA
  • Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
  • Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
Bryan O'Gorman
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA

Acknowledgements

Part of this work was completed while CN and BO were participants in the Simons Institute for the Theory of Computing Summer Cluster on Quantum Computation. Additionally, we thank Sam Gunn, Zeph Landau, Dimitri Maslov and Kristan Temme for insightful discussions.

Cite AsGet BibTex

Srinivasan Arunachalam, Sergey Bravyi, Chinmay Nirkhe, and Bryan O'Gorman. The Parametrized Complexity of Quantum Verification. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 3:1-3:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.TQC.2022.3

Abstract

We initiate the study of parameterized complexity of QMA problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford + t T-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most t qubits (independent of the system size). Furthermore, we derive new lower bounds on the T-count of circuit satisfiability instances and the T-count of the W-state assuming the classical exponential time hypothesis (ETH). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • parametrized complexity
  • quantum verification
  • QMA

Metrics

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

References

  1. Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70(5):052328, 2004. Google Scholar
  2. Scott Aaronson and Greg Kuperberg. Quantum versus classical proofs and advice. In Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07), pages 115-128, 2007. URL: https://doi.org/10.1109/CCC.2007.27.
  3. Dorit Aharonov and Tomer Naveh. Quantum NP-a survey. arXiv quant-ph/0210077, 2002. Google Scholar
  4. F Barahona. On the computational complexity of Ising spin glass models. Journal of Physics A: Mathematical and General, 15(10):3241-3253, October 1982. URL: https://doi.org/10.1088/0305-4470/15/10/028.
  5. Sergei Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71, March 2004. URL: https://doi.org/10.1103/PhysRevA.71.022316.
  6. Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard. Simulation of quantum circuits by low-rank stabilizer decompositions. Quantum, 3:181, September 2019. URL: https://doi.org/10.22331/q-2019-09-02-181.
  7. Sergey Bravyi, David Fattal, and Daniel Gottesman. GHZ extraction yield for multipartite stabilizer states. Journal of Mathematical Physics, 47(6):062106, 2006. Google Scholar
  8. Sergey Bravyi and David Gosset. Improved classical simulation of quantum circuits dominated by Clifford gates. Phys. Rev. Lett., 116:250501, June 2016. URL: https://doi.org/10.1103/PhysRevLett.116.250501.
  9. Sergey Bravyi, Graeme Smith, and John A. Smolin. Trading classical and quantum computational resources. Phys. Rev. X, 6:021043, June 2016. URL: https://doi.org/10.1103/PhysRevX.6.021043.
  10. David Fattal, Toby S Cubitt, Yoshihisa Yamamoto, Sergey Bravyi, and Isaac L Chuang. Entanglement in the stabilizer formalism. arXiv preprint quant-ph/0406168, 2004. Google Scholar
  11. Bill Fefferman and Shelby Kimmel. Quantum vs. classical proofs and subset verification. In 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS, volume 117 of LIPIcs, pages 22:1-22:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  12. Hector J Garcia, Igor L Markov, and Andrew W Cross. Efficient inner-product algorithm for stabilizer states, 2012. arXiv:1210.6646. Google Scholar
  13. Hector J Garcia-Ramirez. Hybrid Techniques for Simulating Quantum Circuits using the Heisenberg Representation. PhD thesis, University of Michigan, 2014. Google Scholar
  14. Daniel Gottesman. The heisenberg representation of quantum computers. Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, 1998. URL: http://arxiv.org/abs/arXiv:quant-ph/9807006.
  15. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, March 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  16. Dominik Janzing, Pawel Wocjan, and Thomas Beth. "non-identity-check" is QMA-complete. International Journal of Quantum Information, 03(03):463-473, 2005. URL: https://doi.org/10.1142/S0219749905001067.
  17. Zhengfeng Ji and Xiaodi Wu. Non-identity check remains QMA-complete for short circuits. arXiv preprint, 2009. URL: http://arxiv.org/abs/0906.5416.
  18. Alexei Yu Kitaev, Alexander Shen, Mikhail N Vyalyi, and Mikhail N Vyalyi. Classical and quantum computation. Number 47 in Graduate Studies in Mathematics. American Mathematical Soc., 2002. Google Scholar
  19. Jacek Kuczyński and Henryk Woźniakowski. Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start. SIAM journal on matrix analysis and applications, 13(4):1094-1122, 1992. Google Scholar
  20. Ashley Montanaro and Ronald de Wolf. A survey of quantum property testing. Theory Comput., 7:1-81, 2016. URL: https://doi.org/10.4086/toc.gs.2016.007.
  21. Tomoyuki Morimae, Masahito Hayashi, Harumichi Nishimura, and Keisuke Fujii. Quantum Merlin-Arthur with Clifford Arthur. arXiv preprint, 2015. URL: http://arxiv.org/abs/1506.06447.
  22. Bryan O'Gorman. Parameterization of Tensor Network Contraction. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019), volume 135 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:19, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  23. Hammam Qassim, Hakop Pashayan, and David Gosset. Improved upper bounds on the stabilizer rank of magic states. arXiv preprint, 2021. URL: http://arxiv.org/abs/2106.07740.
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