Document Open Access Logo

Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States

Authors Chinmay Nirkhe, Umesh Vazirani, Henry Yuen



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2018.91.pdf
  • Filesize: 426 kB
  • 11 pages

Document Identifiers

Author Details

Chinmay Nirkhe
  • Electrical Engineering and Computer Sciences, University of California, Berkeley , 387 Soda Hall Berkeley, CA 94720, U.S.A., https://people.eecs.berkeley.edu/~nirkhe/
Umesh Vazirani
  • Electrical Engineering and Computer Sciences, University of California, Berkeley , 387 Soda Hall Berkeley, CA 94720, U.S.A., https://people.eecs.berkeley.edu/~vazirani/
Henry Yuen
  • Electrical Engineering and Computer Sciences, University of California, Berkeley , 387 Soda Hall Berkeley, CA 94720, U.S.A., https://www.henryyuen.net/

Cite AsGet BibTex

Chinmay Nirkhe, Umesh Vazirani, and Henry Yuen. Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 91:1-91:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.91

Abstract

The No Low-Energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Information and Computation 2014), which asserts the existence of local Hamiltonians whose low-energy states cannot be generated by constant-depth quantum circuits, identifies a fundamental obstacle to resolving the quantum PCP conjecture. Progress towards the NLTS conjecture was made by Eldar and Harrow (Foundations of Computer Science 2017), who proved a closely related theorem called No Low-Error Trivial States (NLETS). In this paper, we give a much simpler proof of the NLETS theorem and use the same technique to establish superpolynomial circuit size lower bounds for noisy ground states of local Hamiltonians (assuming QCMA != QMA), resolving an open question of Eldar and Harrow. We discuss the new light our results cast on the relationship between NLTS and NLETS. Finally, our techniques imply the existence of approximate quantum low-weight check (qLWC) codes with linear rate, linear distance, and constant weight checks. These codes are similar to quantum LDPC codes except (1) each particle may participate in a large number of checks, and (2) errors only need to be corrected up to fidelity 1 - 1/poly(n). This stands in contrast to the best-known stabilizer LDPC codes due to Freedman, Meyer, and Luo which achieve a distance of O(sqrt{n log n}). The principal technique used in our results is to leverage the Feynman-Kitaev clock construction to approximately embed a subspace of states defined by a circuit as the ground space of a local Hamiltonian.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum pcps
  • local hamiltonians
  • error-correcting codes

Metrics

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

References

  1. Scott Aaronson and Greg Kuperberg. Quantum versus classical proofs and advice. In Computational Complexity, 2007. CCC'07. Twenty-Second Annual IEEE Conference on, pages 115-128. IEEE, 2007. Google Scholar
  2. Dorit Aharonov, Itai Arad, and Thomas Vidick. Guest column: The quantum PCP conjecture. SIGACT News, 44(2):47-79, 2013. URL: http://dx.doi.org/10.1145/2491533.2491549.
  3. Dorit Aharonov and Lior Eldar. Quantum locally testable codes. SIAM Journal on Computing, 44(5):1230-1262, 2015. URL: http://dx.doi.org/10.1137/140975498.
  4. Dorit Aharonov and Tomer Naveh. Quantum NP - a survey, 2002. URL: http://arxiv.org/abs/arXiv:quant-ph/0210077.
  5. Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J. Comput., 37(1):166-194, 2007. URL: http://dx.doi.org/10.1137/S0097539705447323.
  6. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: http://dx.doi.org/10.1145/278298.278306.
  7. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of np. J. ACM, 45(1):70-122, 1998. URL: http://dx.doi.org/10.1145/273865.273901.
  8. Alexei Ashikhmin, Simon Litsyn, and Michael A Tsfasman. Asymptotically good quantum codes. Physical Review A, 63(3):032311, 2001. Google Scholar
  9. Dave Bacon, Steven T Flammia, Aram W Harrow, and Jonathan Shi. Sparse quantum codes from quantum circuits. IEEE Transactions on Information Theory, 63(4):2464-2479, 2017. Google Scholar
  10. Johannes Bausch and Elizabeth Crosson. Analysis and limitations of modified circuit-to-hamiltonian constructions, 2016. URL: http://arxiv.org/abs/arXiv:1609.08571.
  11. Cédric Bény and Ognyan Oreshkov. General conditions for approximate quantum error correction and near-optimal recovery channels. Physical review letters, 104(12):120501, 2010. Google Scholar
  12. Fernando GSL Brandao, Elizabeth Crosson, M Burak Şahinoğlu, and John Bowen. Quantum error correcting codes in eigenstates of translation-invariant spin chains. arXiv preprint arXiv:1710.04631, 2017. Google Scholar
  13. Sergey Bravyi, David Poulin, and Barbara Terhal. Tradeoffs for reliable quantum information storage in 2d systems. Physical review letters, 104(5):050503, 2010. Google Scholar
  14. Libor Caha, Zeph Landau, and Daniel Nagaj. The Feynman-Kitaev computer’s clock: bias, gaps, idling and pulse tuning. arXiv preprint arXiv:1712.07395, 2017. Google Scholar
  15. Stephen A. Cook. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC '71, pages 151-158, New York, NY, USA, 1971. ACM. URL: http://dx.doi.org/10.1145/800157.805047.
  16. Claude Crépeau, Daniel Gottesman, and Adam Smith. Approximate quantum error-correcting codes and secret sharing schemes. In Proceedings of the 24th Annual International Conference on Theory and Applications of Cryptographic Techniques, EUROCRYPT'05, pages 285-301, Berlin, Heidelberg, 2005. Springer-Verlag. URL: http://dx.doi.org/10.1007/11426639_17.
  17. Elizabeth Crosson and John F Bowen. Quantum ground state isoperimetric inequalities for the energy spectrum of local hamiltonians. arXiv quant-ph, 2017. URL: http://arxiv.org/abs/1703.10133.
  18. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3), jun 2007. URL: http://dx.doi.org/10.1145/1236457.1236459.
  19. Lior Eldar and Aram Wettroth Harrow. Local hamiltonians whose ground states are hard to approximate. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 427-438, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.46.
  20. Bill Fefferman and Shelby Kimmel. Quantum vs classical proofs and subset verification. CoRR, abs/1510.06750, 2015. URL: http://arxiv.org/abs/1510.06750.
  21. Steven T. Flammia, Jeongwan Haah, Michael J. Kastoryano, and Isaac H. Kim. Limits on the storage of quantum information in a volume of space. Quantum, 1:4, apr 2017. URL: http://dx.doi.org/10.22331/q-2017-04-25-4.
  22. Michael H. Freedman and Matthew B. Hastings. Quantum systems on non-k-hyperfinite complexes: a generalization of classical statistical mechanics on expander graphs. Quantum Information & Computation, 14(1-2):144-180, 2014. URL: http://www.rintonpress.com/xxqic14/qic-14-12/0144-0180.pdf.
  23. Michael H Freedman, David A Meyer, and Feng Luo. Z2-systolic freedom and quantum codes. Mathematics of quantum computation, Chapman &Hall/CRC, pages 287-320, 2002. Google Scholar
  24. Daniel Eric. Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, Pasadena, California, 1997. Google Scholar
  25. Matthew B Hastings. Weight reduction for quantum codes. arXiv preprint arXiv:1611.03790, 2016. Google Scholar
  26. Matthew B. Hastings. Quantum codes from high-dimensional manifolds. In 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, pages 25:1-25:26, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2017.25.
  27. Patrick Hayden and Geoffrey Penington. Approximate quantum error correction revisited: Introducing the alphabit. arXiv preprint arXiv:1706.09434, 2017. Google Scholar
  28. Isaac H Kim and Michael J Kastoryano. Entanglement renormalization, quantum error correction, and bulk causality. Journal of High Energy Physics, 2017(4):40, 2017. Google Scholar
  29. A.Y. Kitaev, A. Shen, and M.N. Vyalyi. Classical and Quantum Computation. Graduate studies in mathematics. American Mathematical Society, 2002. Google Scholar
  30. A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2-30, 2003. URL: http://dx.doi.org/10.1016/S0003-4916(02)00018-0.
  31. Debbie W Leung, Michael A Nielsen, Isaac L Chuang, and Yoshihisa Yamamoto. Approximate quantum error correction can lead to better codes. Physical Review A, 56(4):2567, 1997. Google Scholar
  32. L. A. Levin. Universal sequential search problems. Problems of Information Transmission, 9(3):265-266, 1973. Google Scholar
  33. Jean-Pierre Tillich and Gilles Zémor. Quantum ldpc codes with positive rate and minimum distance proportional to the square root of the blocklength. IEEE Transactions on Information Theory, 60(2):1193-1202, 2014. Google Scholar
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