Document Open Access Logo

Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians

Authors Anurag Anshu , Chinmay Nirkhe

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2022.6.pdf
  • Filesize: 0.84 MB
  • 22 pages

Document Identifiers

Author Details

Anurag Anshu
  • Simons Institute for the Theory of Computing, Berkeley, California, USA
  • Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
  • Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
Chinmay Nirkhe
  • Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
  • Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA

Funding

  • Anshu, Anurag: was supported by NSF Quantum Leap Challenges Institute Grant number OMA2016245 and part of this work was completed while he was a participant in the Simons Institute for the Theory of Computing program on The Quantum Wave in Computing.
  • Nirkhe, Chinmay: was supported by NSF Quantum Leap Challenges Institute Grant number OMA2016245 and part of this work was completed while he was a participant in the Simons Institute for the Theory of Computing program on The Quantum Wave in Computing. He also acknowledges support from an IBM Quantum PhD Internship.

Acknowledgements

We thank Lior Eldar, Zeph Landau, Umesh Vazirani, and anonymous conference reviewers for detailed comments on the manuscript that greatly improved the presentation. Additional thanks to Dorit Aharanov, Matt Hastings, Aleksander Kubica, Rishabh Pipada, Elizabeth Yang, and Henry Yuen for helpful discussions.

Cite AsGet BibTex

Anurag Anshu and Chinmay Nirkhe. Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 6:1-6:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.6

Abstract

The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings [Freedman and Hastings, 2014] - which posits the existence of a local Hamiltonian with a super-constant quantum circuit lower bound on the complexity of all low-energy states - identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes. For local Hamiltonians arising from nearly linear-rate or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). Such codes are known to exist and are not necessarily locally-testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems.

Subject Classification

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

Metrics

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

Related Versions

References

  1. Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70:052328, November 2004. URL: https://doi.org/10.1103/PhysRevA.70.052328.
  2. Dorit Aharonov, Itai Arad, Zeph Landau, and Umesh Vazirani. The detectability lemma and quantum gap amplification. In Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, STOC '09, pages 417-426, New York, NY, USA, 2009. Association for Computing Machinery. URL: https://doi.org/10.1145/1536414.1536472.
  3. Dorit Aharonov, Itai Arad, and Thomas Vidick. Guest column: The quantum pcp conjecture. SIGACT News, 44(2):47-79, June 2013. URL: https://doi.org/10.1145/2491533.2491549.
  4. Dorit Aharonov and Lior Eldar. The commuting local hamiltonian problem on locally expanding graphs is approximable in np. Quantum Information Processing, 14(1):83-101, January 2015. URL: https://doi.org/10.1007/s11128-014-0877-9.
  5. Dorit Aharonov and Lior Eldar. Quantum locally testable codes. SIAM Journal on Computing, 44(5):1230-1262, 2015. URL: https://doi.org/10.1137/140975498.
  6. Dorit Aharonov and Tomer Naveh. Quantum np - a survey, 2002. URL: http://arxiv.org/abs/quant-ph/0210077.
  7. Anurag Anshu, Aram W. Harrow, and Mehdi Soleimanifar. From communication complexity to an entanglement spread area law in the ground state of gapped local hamiltonians. arXiv, 2020. URL: http://arxiv.org/abs/2004.15009.
  8. Anurag Anshu and Chinmay Nirkhe. Circuit lower bounds for low-energy states of quantum code hamiltonians, 2020. URL: http://arxiv.org/abs/2011.02044.
  9. B. Apolloni, C. Carvalho, and D. de Falco. Quantum stochastic optimization. Stochastic Processes and their Applications, 33(2):233-244, 1989. URL: https://doi.org/10.1016/0304-4149(89)90040-9.
  10. Itai Arad, Alexei Kitaev, Zeph Landau, and Umesh Vazirani. An area law and sub-exponential algorithm for 1D systems, 2013. URL: http://arxiv.org/abs/1301.1162.
  11. Itai Arad, Zeph Landau, and Umesh Vazirani. Improved one-dimensional area law for frustration-free systems. Physical Review B, 85:195145, May 2012. URL: https://doi.org/10.1103/PhysRevB.85.195145.
  12. Johannes Bausch and Elizabeth Crosson. Analysis and limitations of modified circuit-to-Hamiltonian constructions. Quantum, 2:94, September 2018. URL: https://doi.org/10.22331/q-2018-09-19-94.
  13. Chris Beck, Russell Impagliazzo, and Shachar Lovett. Large deviation bounds for decision trees and sampling lower bounds for ac0-circuits. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 101-110, 2012. URL: https://doi.org/10.1109/FOCS.2012.82.
  14. Fernando G.S.L. Brandao and Aram W. Harrow. Product-state approximations to quantum ground states. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 871-880, New York, NY, USA, 2013. Association for Computing Machinery. URL: https://doi.org/10.1145/2488608.2488719.
  15. Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang. Obstacles to state preparation and variational optimization from symmetry protection. arXiv, 2019. URL: http://arxiv.org/abs/1910.08980.
  16. Sergey Bravyi, David Poulin, and Barbara Terhal. Tradeoffs for reliable quantum information storage in 2d systems. Phys. Rev. Lett., 104:050503, February 2010. URL: https://doi.org/10.1103/PhysRevLett.104.050503.
  17. Sergey Bravyi and Mikhail Vyalyi. Commutative version of the local hamiltonian problem and common eigenspace problem. Quantum Info. Comput., 5(3):187-215, May 2005. Google Scholar
  18. N. P. Breuckmann and J. Eberhardt. Balanced product quantum codes. arXiv: Quantum Physics, 2020. Google Scholar
  19. Harry Buhrman, Richard Cleve, Ronald de Wolf, and Christof Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, FOCS '99, page 358, USA, 1999. IEEE Computer Society. Google Scholar
  20. Libor Caha, Zeph Landau, and Daniel Nagaj. Clocks in feynman’s computer and kitaev’s local hamiltonian: Bias, gaps, idling, and pulse tuning. Phys. Rev. A, 97:062306, June 2018. URL: https://doi.org/10.1103/PhysRevA.97.062306.
  21. Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik. Quantum chemistry in the age of quantum computing. Chemical Reviews, 119(19):10856-10915, October 2019. URL: https://doi.org/10.1021/acs.chemrev.8b00803.
  22. Irit Dinur. The pcp theorem by gap amplification. J. ACM, 54(3):12-es, June 2007. URL: https://doi.org/10.1145/1236457.1236459.
  23. L. Eldar and A. W. Harrow. Local hamiltonians whose ground states are hard to approximate. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 427-438, 2017. URL: https://doi.org/10.1109/FOCS.2017.46.
  24. Lior Eldar. Robust quantum entanglement at (nearly) room temperature. arXiv: Quantum Physics (to appear in ITCS 2021), 2019. Google Scholar
  25. Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint, 2014. URL: http://arxiv.org/abs/1411.4028.
  26. Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86:032324, September 2012. URL: https://doi.org/10.1103/PhysRevA.86.032324.
  27. Michael H. Freedman and Matthew B. Hastings. Quantum systems on non-k-hyperfinite complexes: A generalization of classical statistical mechanics on expander graphs. Quantum Info. Comput., 14(1–2):144-180, January 2014. Google Scholar
  28. Larry Guth and Alexander Lubotzky. Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds. Journal of Mathematical Physics, 55(8):082202, 2014. URL: https://doi.org/10.1063/1.4891487.
  29. Matthew B Hastings. An area law for one-dimensional quantum systems. Journal of Statistical Mechanics: Theory and Experiment, 2007(08):P08024, 2007. Google Scholar
  30. Matthew B. Hastings. Topological order at nonzero temperature. Phys. Rev. Lett., 107:210501, November 2011. URL: https://doi.org/10.1103/PhysRevLett.107.210501.
  31. Matthew B. Hastings. Quantum Codes from High-Dimensional Manifolds. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pages 25:1-25:26, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.25.
  32. Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell. Fiber bundle codes: Breaking the n^1/2 polylog(n) barrier for quantum ldpc codes, 2020. URL: http://arxiv.org/abs/2009.03921.
  33. Tadashi Kadowaki and Hidetoshi Nishimori. Quantum annealing in the transverse ising model. Phys. Rev. E, 58:5355-5363, November 1998. URL: https://doi.org/10.1103/PhysRevE.58.5355.
  34. Jeff Kahn, Nathan Linial, and Alex Samorodnitsky. Inclusion-exclusion: Exact and approximate. Combinatorica, 16(4):465-477, December 1996. URL: https://doi.org/10.1007/BF01271266.
  35. Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local hamiltonian problem. SIAM Journal on Computing, 35(5):1070-1097, 2006. URL: https://doi.org/10.1137/S0097539704445226.
  36. A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. Classical and Quantum Computation. American Mathematical Society, USA, 2002. Google Scholar
  37. Alexei Kitaev and John Preskill. Topological entanglement entropy. Phys. Rev. Lett., 96:110404, March 2006. URL: https://doi.org/10.1103/PhysRevLett.96.110404.
  38. A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2-30, 2003. URL: https://doi.org/10.1016/S0003-4916(02)00018-0.
  39. Emanuel Knill and Raymond Laflamme. Theory of quantum error-correcting codes. Phys. Rev. A, 55:900-911, February 1997. URL: https://doi.org/10.1103/PhysRevA.55.900.
  40. Ryan LaRose, Arkin Tikku, Étude O'Neel-Judy, Lukasz Cincio, and Patrick J. Coles. Variational quantum state diagonalization. npj Quantum Information, 5(1):57, June 2019. URL: https://doi.org/10.1038/s41534-019-0167-6.
  41. Anthony Leverrier, Vivien Londe, and Gilles Zémor. Towards local testability for quantum coding, 2019. URL: http://arxiv.org/abs/1911.03069.
  42. Vivien Londe and Anthony Leverrier. Golden codes: Quantum ldpc codes built from regular tessellations of hyperbolic 4-manifolds. Quantum Info. Comput., 19(5–6):361-391, May 2019. Google Scholar
  43. Shachar Lovett and Emanuele Viola. Bounded-depth circuits cannot sample good codes. Electronic Colloquium on Computational Complexity (ECCC), 17:115, October 2010. URL: https://doi.org/10.1007/s00037-012-0039-3.
  44. Anand Natarajan and Thomas Vidick. Low-degree testing for quantum states, and a quantum entangled games PCP for QMA. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 731-742. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00075.
  45. Chinmay Nirkhe, Umesh Vazirani, and Henry Yuen. Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 91:1-91:11, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.91.
  46. Pavel Panteleev and Gleb Kalachev. Quantum ldpc codes with almost linear minimum distance, 2020. URL: http://arxiv.org/abs/2012.04068.
  47. Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5(1):4213, July 2014. URL: https://doi.org/10.1038/ncomms5213.
  48. Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett., 86:5188-5191, May 2001. URL: https://doi.org/10.1103/PhysRevLett.86.5188.
  49. Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Phys. Rev. A, 68:022312, August 2003. URL: https://doi.org/10.1103/PhysRevA.68.022312.
  50. J. Tillich and G. Zemor. Quantum ldpc codes with positive rate and minimum distance proportional to sqrt(n). In 2009 IEEE International Symposium on Information Theory, pages 799-803, 2009. URL: https://doi.org/10.1109/ISIT.2009.5205648.
  51. A. Uhlmann. The "transition probability" in the state space of a *-algebra. Rep. Math. Phys., 9:273-279, 1976. Google Scholar
  52. Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69:2863-2866, November 1992. URL: https://doi.org/10.1103/PhysRevLett.69.2863.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact