Document Open Access Logo

A Qubit, a Coin, and an Advice String Walk into a Relational Problem

Authors Scott Aaronson, Harry Buhrman, William Kretschmer

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2024.1.pdf
  • Filesize: 0.83 MB
  • 24 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Quantum complexity theory
Keywords
  • Relational problems
  • quantum advice
  • randomized advice
  • FBQP
  • FBPP

Metrics

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

Abstract

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly).
Our first result is that FBQP/qpoly ≠ FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions.
Our second result is that FBPP ̸ ⊂ FP/poly - that is, Adleman’s Theorem fails for relational problems - unless PSPACE ⊂ NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP ̸ ⊂ FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP.
We prove the following further results:  
- Unconditionally, FP ≠ FBPP and FP/poly ≠ FBPP/poly (even when these classes are carefully defined). 
- FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly ≠ SampBPP/rpoly (and likewise for SampBQP).

Cite As Get BibTex

Scott Aaronson, Harry Buhrman, and William Kretschmer. A Qubit, a Coin, and an Advice String Walk into a Relational Problem. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.1

Author Details

Scott Aaronson
  • University of Texas at Austin, TX, USA
  • OpenAI, San Francisco, CA, USA
Harry Buhrman
  • QuSoft, Amsterdam, The Netherlands
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands
William Kretschmer
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
  • University of California, Berkeley, CA, USA

Funding

  • Aaronson, Scott: Supported by a Vannevar Bush Fellowship from the US Department of Defense, the Berkeley NSF-QLCI CIQC Center, a Simons Investigator Award, and the Simons quotedblleft It from Qubitquotedblright collaboration.
  • Buhrman, Harry: Supported in part by the Dutch Research Council (NWO/OCW), Gravitation Programmes Quantum Software Consortium (project number 024.003.037) and Networks (project number 024.002.003).
  • Kretschmer, William: Supported by an NDSEG Fellowship and a Simons Quantum Postdoctoral Fellowship. Support is also acknowledged from the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator.

Acknowledgements

We thank Daochen Wang and Alexandru Cojocaru for conversations that suggested the FBQP/poly versus FBQP/qpoly problem, and Lance Fortnow, Iordanis Kerenidis, Ashley Montanaro, Ronald de Wolf, and David Zuckerman for helpful discussions. We thank anonymous referees for constructive feedback.

References

  1. Scott Aaronson. Limitations of quantum advice and one-way communication. Theory of Computing, 1(1):1-28, 2005. URL: https://doi.org/10.4086/toc.2005.v001a001.
  2. Scott Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society A, 461:3473-3482, 2005. URL: https://doi.org/10.1098/rspa.2005.1546.
  3. Scott Aaronson. BQP and the polynomial hierarchy. In Proceedings of the Forty-Second ACM Symposium on Theory of Computing, STOC '10, pages 141-150, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1806689.1806711.
  4. Scott Aaronson. The equivalence of sampling and searching. Theory of Computing Systems, 55(2):281-298, 2014. URL: https://doi.org/10.1007/s00224-013-9527-3.
  5. Scott Aaronson and Andris Ambainis. The need for structure in quantum speedups. Theory of Computing, 10(6):133-166, 2014. URL: https://doi.org/10.4086/toc.2014.v010a006.
  6. Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. Theory of Computing, 9(4):143-252, 2013. URL: https://doi.org/10.4086/toc.2013.v009a004.
  7. Scott Aaronson and Lijie Chen. Complexity-Theoretic Foundations of Quantum Supremacy Experiments. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference (CCC 2017), volume 79 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:67, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.22.
  8. Scott Aaronson and Sam Gunn. On the classical hardness of spoofing linear cross-entropy benchmarking. Theory of Computing, 16(11):1-8, 2020. URL: https://doi.org/10.4086/toc.2020.v016a011.
  9. Scott Aaronson and Greg Kuperberg. Quantum versus classical proofs and advice. Theory of Computing, 3(7):129-157, 2007. URL: https://doi.org/10.4086/toc.2007.v003a007.
  10. Leonard Adleman. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science (SFCS 1978), pages 75-83, 1978. URL: https://doi.org/10.1109/SFCS.1978.37.
  11. Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. A polynomial-time classical algorithm for noisy random circuit sampling. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 945-957, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585234.
  12. Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505-510, 2019. URL: https://doi.org/10.1038/s41586-019-1666-5.
  13. Ziv Bar-Yossef, T. S. Jayram, and Iordanis Kerenidis. Exponential separation of quantum and classical one-way communication complexity. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC '04, pages 128-137, New York, NY, USA, 2004. Association for Computing Machinery. URL: https://doi.org/10.1145/1007352.1007379.
  14. Elwyn R. Berlekamp. Factoring polynomials over large finite fields. Mathematics of Computation, 24(111):713-735, 1970. URL: https://doi.org/10.2307/2004849.
  15. Harry Buhrman and Leen Torenvliet. Randomness is hard. SIAM Journal on Computing, 30(5):1485-1501, 2000. URL: https://doi.org/10.1137/S0097539799360148.
  16. Bill Fefferman and Shelby Kimmel. Quantum vs. Classical Proofs and Subset Verification. In Igor Potapov, Paul Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), volume 117 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:23, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.22.
  17. Xun Gao, Marcin Kalinowski, Chi-Ning Chou, Mikhail D. Lukin, Boaz Barak, and Soonwon Choi. Limitations of linear cross-entropy as a measure for quantum advantage, 2021. URL: https://arxiv.org/abs/2112.01657.
  18. Dmitry Gavinsky, Julia Kempe, Iordanis Kerenidis, Ran Raz, and Ronald de Wolf. Exponential separation for one-way quantum communication complexity, with applications to cryptography. SIAM Journal on Computing, 38(5):1695-1708, 2009. URL: https://doi.org/10.1137/070706550.
  19. Oded Goldreich. In a World of P=BPP, pages 191-232. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_20.
  20. David Gosset and John Smolin. A Compressed Classical Description of Quantum States. In Wim van Dam and Laura Mančinska, editors, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019), volume 135 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:9, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2019.8.
  21. Alexander S. Holevo. Bounds for the quantity of information transmitted by a quantum communication channel. Problems Inform. Transmission, 9(3):177-183, 1973. Translated from Russian. Google Scholar
  22. Rahul Ilango, Jiatu Li, and R. Ryan Williams. Indistinguishability obfuscation, range avoidance, and bounded arithmetic. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1076-1089, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585187.
  23. Richard M. Karp and RIchard J. Lipton. Turing machines that take advice. L'Enseignement Mathématique, 28:191-209, 1982. URL: https://doi.org/10.5169/seals-52237.
  24. Niraj Kumar, Iordanis Kerenidis, and Eleni Diamanti. Experimental demonstration of quantum advantage for one-way communication complexity surpassing best-known classical protocol. Nature Communications, 10(1):4152, 2019. URL: https://doi.org/10.1038/s41467-019-12139-z.
  25. Ryan L. Mann. Quantum computation and combinatorial structures. PhD thesis, University of Technology Sydney, February 2019. URL: https://doi.org/10453/133334.
  26. Ashley Montanaro. Quantum states cannot be transmitted efficiently classically. Quantum, 3:154, June 2019. URL: https://doi.org/10.22331/q-2019-06-28-154.
  27. Tomoyuki Morimae and Takashi Yamakawa. Quantum advantage from one-way functions, 2023. URL: https://arxiv.org/abs/2302.04749.
  28. Anand Natarajan and Chinmay Nirkhe. A Distribution Testing Oracle Separating QMA and QCMA. In Amnon Ta-Shma, editor, 38th Computational Complexity Conference (CCC 2023), volume 264 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:27, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.22.
  29. Harumichi Nishimura and Tomoyuki Yamakami. Polynomial time quantum computation with advice. Information Processing Letters, 90(4):195-204, 2004. URL: https://doi.org/10.1016/j.ipl.2004.02.005.
  30. OpenAI. GPT-4 technical report, 2023. URL: https://arxiv.org/abs/2303.08774.
  31. Feng Pan, Keyang Chen, and Pan Zhang. Solving the sampling problem of the sycamore quantum circuits. Phys. Rev. Lett., 129:090502, August 2022. URL: https://doi.org/10.1103/PhysRevLett.129.090502.
  32. Adi Shamir. IP = PSPACE. J. ACM, 39(4):869-877, October 1992. URL: https://doi.org/10.1145/146585.146609.
  33. Larry Stockmeyer. On approximation algorithms for #P. SIAM Journal on Computing, 14(4):849-861, 1985. URL: https://doi.org/10.1137/0214060.
  34. Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, Cheng Guo, Chu Guo, Shaojun Guo, Lianchen Han, Linyin Hong, He-Liang Huang, Yong-Heng Huo, Liping Li, Na Li, Shaowei Li, Yuan Li, Futian Liang, Chun Lin, Jin Lin, Haoran Qian, Dan Qiao, Hao Rong, Hong Su, Lihua Sun, Liangyuan Wang, Shiyu Wang, Dachao Wu, Yu Xu, Kai Yan, Weifeng Yang, Yang Yang, Yangsen Ye, Jianghan Yin, Chong Ying, Jiale Yu, Chen Zha, Cha Zhang, Haibin Zhang, Kaili Zhang, Yiming Zhang, Han Zhao, Youwei Zhao, Liang Zhou, Qingling Zhu, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, and Jian-Wei Pan. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett., 127:180501, October 2021. URL: https://doi.org/10.1103/PhysRevLett.127.180501.
  35. Takashi Yamakawa and Mark Zhandry. Verifiable quantum advantage without structure. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 69-74, Los Alamitos, CA, USA, November 2022. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS54457.2022.00014.
  36. Andrew Chi-Chih Yao. Probabilistic computations: Toward a unified measure of complexity. In 18th Annual Symposium on Foundations of Computer Science (SFCS 1977), pages 222-227, 1977. URL: https://doi.org/10.1109/SFCS.1977.24.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact