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The Complexity of Translationally Invariant Problems Beyond Ground State Energies

Authors James D. Watson , Johannes Bausch , Sevag Gharibian

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Author Details

James D. Watson
  • University College London, UK
Johannes Bausch
  • University of Cambridge, UK
Sevag Gharibian
  • Universität Paderborn, Germany

Funding

  • Watson, James D.: EPSRC Centre for Doctoral Training in Delivering Quantum Technologies (grant EP/L015242/1).
  • Bausch, Johannes: Draper’s Research Fellowship at Pembroke College.
  • Gharibian, Sevag: DFG grant 432788384

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James D. Watson, Johannes Bausch, and Sevag Gharibian. The Complexity of Translationally Invariant Problems Beyond Ground State Energies. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 54:1-54:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.54

Abstract

The physically motivated quantum generalisation of k-SAT, the k-Local Hamiltonian (k-LH) problem, is well-known to be QMA-complete ("quantum NP"-complete). What is surprising, however, is that while the former is easy on 1D Boolean formulae, the latter remains hard on 1D local Hamiltonians, even if all constraints are identical [Gottesman, Irani, FOCS 2009]. Such "translation-invariant" systems are much closer in structure to what one might see in Nature. Moving beyond k-LH, what is often more physically interesting is the computation of properties of the ground space (i.e. "solution space") itself. In this work, we focus on two such recent problems: Simulating local measurements on the ground space (APX-SIM, analogous to computing properties of optimal solutions to MAX-SAT formulae) [Ambainis, CCC 2014], and deciding if the low energy space has an energy barrier (GSCON, analogous to classical reconfiguration problems) [Gharibian, Sikora, ICALP 2015]. These problems are known to be P^{QMA[log]}- and QCMA-complete, respectively, in the general case. Yet, to date, it is not known whether they remain hard in such simple 1D translationally invariant systems. In this work, we show that the 1D translationally invariant versions of both APX-SIM and GSCON are intractable, namely are P^{QMA_{EXP}}- and QCMA^{EXP}-complete ("quantum P^{NEXP}" and "quantum NEXP"), respectively. Each of these results is attained by giving a respective generic "lifting theorem". For APX-SIM we give a framework for lifting any abstract local circuit-to-Hamiltonian mapping H satisfying mild assumptions to hardness of APX-SIM on the family of Hamiltonians produced by H, while preserving the structural properties of H (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we compress the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we show strong robustness, i.e. soundness even against adversaries acting on all but a single qudit in the system.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Quantum complexity theory
Keywords
  • Complexity
  • Quantum Computing
  • Physics
  • Constraint Satisfaction
  • Combinatorial Reconfiguration
  • Many-Body Physics

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References

  1. Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Rigorous results on valence-bond ground states in antiferromagnets. Physical Review Letters, 59(7):799-802, August 1987. URL: https://doi.org/10.1103/PhysRevLett.59.799.
  2. Dorit Aharonov, Daniel Gottesman, Sandy Irani, and Julia Kempe. The power of quantum systems on a line. Communications in Mathematical Physics, 287(1):41-65, May 2009. URL: https://doi.org/10.1007/s00220-008-0710-3.
  3. Andris Ambainis. On physical problems that are slightly more difficult than QMA. In 29th IEEE Conference on Computational Complexity (CCC), pages 32-43, 2014. Google Scholar
  4. Johannes Bausch, Toby Cubitt, and Maris Ozols. The Complexity of Translationally Invariant Spin Chains with Low Local Dimension. Annales Henri Poincaré, 18(11):3449-3513, November 2017. URL: https://doi.org/10.1007/s00023-017-0609-7.
  5. Johannes Bausch, Toby S. Cubitt, Angelo Lucia, and David Perez-Garcia. Undecidability of the Spectral Gap in One Dimension. Physical Review X, 10(3):031038, August 2020. URL: https://doi.org/10.1103/PhysRevX.10.031038.
  6. Johannes Bausch and Stephen Piddock. The complexity of translationally invariant low-dimensional spin lattices in 3D. Journal of Mathematical Physics, 58(11):111901, November 2017. URL: https://doi.org/10.1063/1.5011338.
  7. H Bethe. Zur Theorie der Metalle. Zeitschrift für Physik, 71(3-4):205-226, 1931. Google Scholar
  8. S. Bravyi and M. Hastings. On complexity of the quantum Ising model. Communications in Mathematical Physics, 349(1):1-45, 2017. URL: https://doi.org/10.1007/s00220-016-2787-4.
  9. Nikolas P. Breuckmann and Barbara M. Terhal. Space-time circuit-to-Hamiltonian construction and its applications. Journal of Physics A Mathematical General, 47(19):195304, May 2014. URL: https://doi.org/10.1088/1751-8113/47/19/195304.
  10. Brielin Brown, Steven T. Flammia, and Norbert Schuch. Computational Difficulty of Computing the Density of States. PRL, 107(4):040501, July 2011. URL: https://doi.org/10.1103/PhysRevLett.107.040501.
  11. Toby S Cubitt, Ashley Montanaro, and Stephen Piddock. Universal quantum hamiltonians. Proceedings of the National Academy of Sciences, 115(38):9497-9502, 2018. URL: http://arxiv.org/abs/1701.05182.
  12. Toby S. Cubitt, David Perez-Garcia, and Michael M. Wolf. Undecidability of the spectral gap. Nature, 528(7581):207-211, December 2015. URL: https://doi.org/10.1038/nature16059.
  13. Bill Fefferman and Cedric Yen-Yu Lin. A Complete Characterization of Unitary Quantum Space. In Leibniz International Proceedings in Informatics (LIPIcs). 9th Innovations in Theoretical Computer Science Conference, April 2016. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.4.
  14. Fabio Franchini. An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, volume 940 of Lecture Notes in Physics. Springer International Publishing, Cham, 2017. URL: https://doi.org/10.1007/978-3-319-48487-7.
  15. S. Gharibian, Y. Huang, Z. Landau, and S. W. Shin. Quantum Hamiltonian complexity. Foundations and Trends® in Theoretical Computer Science, 10(3):159-282, 2014. URL: https://doi.org/10.1561/0400000066.
  16. S. Gharibian and J. Kempe. Hardness of approximation for quantum problems. In 39th International Colloquium on Automata, Languages and Programming (ICALP), pages 387-398, 2012. Google Scholar
  17. Sevag Gharibian, Stephen Piddock, and Justin Yirka. Oracle Complexity Classes and Local Measurements on Physical Hamiltonians. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020), volume 154, pages 20:1-20:37, 2020. Google Scholar
  18. Sevag Gharibian and Jamie Sikora. Ground state connectivity of local Hamiltonians. In 42nd International Colloquium on Automata, Languages, and Programming (ICALP), pages 617-628, 2015. Google Scholar
  19. Sevag Gharibian and Justin Yirka. The complexity of simulating local measurements on quantum systems. Quantum, 3:189, 2019. URL: https://doi.org/10.22331/q-2019-09-30-189.
  20. David Gosset, Jenish C. Mehta, and Thomas Vidick. QCMA hardness of ground space connectivity for commuting Hamiltonians. Quantum, 1:16, July 2017. URL: https://doi.org/10.22331/q-2017-07-14-16.
  21. Daniel Gottesman and Sandy Irani. The Quantum and Classical Complexity of Translationally Invariant Tiling and Hamiltonian Problems. Theory of Computing, 9(1):31-116, May 2009. URL: https://doi.org/10.4086/toc.2013.v009a002.
  22. S. Hallgren, D. Nagaj, and S. Narazanaswami. The Local Hamiltonian problem on a line with eight states is QMA-complete. Quantum Information & Computation, 13(9&10):0721-0750, 2013. Google Scholar
  23. A. S. Holevo. Bounds for the quantity of information transmitted by a quantum communication channel. Problems of Information Transmission, 9(3):177-183, 1973. Google Scholar
  24. Julia Kempe, Alexei Yu. Kitaev, and Oded Regev. The Complexity of the Local Hamiltonian Problem. SIAM Journal on Computing, 35(5):1070-1097, January 2006. URL: https://doi.org/10.1137/S0097539704445226.
  25. A. Kitaev, A. Shen, and M. Vyalyi. Classical and Quantum Computation. American Mathematical Society, 2002. Google Scholar
  26. Alexei Yu. Kitaev, Alexander Shen, and Mikhail N. Vyalyi. Classical and quantum computing. In Quantum Information, pages 203-217. Springer New York, New York, NY, 2002. URL: https://doi.org/10.1007/978-0-387-36944-0_13.
  27. Tamara Kohler, Stephen Piddock, Johannes Bausch, and Toby Cubitt. Translationally invariant universal quantum hamiltonians in 1d. Annales Henri Poincaré, 23(1):223-254, 2021. URL: https://doi.org/10.1007/s00023-021-01111-7.
  28. Zeph Landau, Umesh Vazirani, and Thomas Vidick. A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians. Nature Physics, 11(7):566-569, June 2015. URL: https://doi.org/10.1038/nphys3345.
  29. Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3):407-466, December 1961. URL: https://doi.org/10.1016/0003-4916(61)90115-4.
  30. Daniel Nagaj. Local Hamiltonians in Quantum Computation. PhD thesis, Massachusetts Institute of Technology, Boston, 2008. Available at arXiv.org quant-ph/0808.2117v1. Google Scholar
  31. Roberto I. Oliveira and Barbara M. Terhal. The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Information and Computation, 8(10):1-23, April 2005. URL: http://arxiv.org/abs/0504050.
  32. T. J. Osborne. Hamiltonian complexity. Reports on Progress in Physics, 75(2):022001, 2012. URL: http://stacks.iop.org/0034-4885/75/i=2/a=022001.
  33. Stephen Piddock and Johannes Bausch. Universal Translationally-Invariant Hamiltonians. arXiv, January 2020. URL: http://arxiv.org/abs/2001.08050.
  34. Y. Shi and S. Zhang. Note on quantum counting classes. Available at URL: http://www.cse.cuhk.edu.hk/syzhang/papers/SharpBQP.pdf.
  35. James D. Watson, Johannes Bausch, and Sevag Gharibian. The Complexity of Translationally Invariant Problems beyond Ground State Energies. arXiv e-prints, page arXiv:2012.12717, December 2020. URL: http://arxiv.org/abs/2012.12717.
  36. Steven R. White. Density matrix formulation for quantum renormalization groups. Physical Review Letters, 69(19):2863-2866, November 1992. URL: https://doi.org/10.1103/PhysRevLett.69.2863.
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