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Undecidability of the spectral gap

Nature volume 528, pages 207211 (10 December 2015) | Download Citation

Abstract

The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

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Acknowledgements

T.S.C. thanks IBM. T. J. Watson Laboratory for their hospitality, and C. Bennett in particular for discussions about this work. T.S.C., D.P.-G. and M.M.W. thank the Isaac Newton Institute for Mathematical Sciences, Cambridge for their hospitality during the programme “Mathematical Challenges in Quantum Information”, where part of this work was carried out. T.S.C. is supported by the Royal Society. D.P.G. acknowledges support from MINECO (grant MTM2011-26912 and PRI-PIMCHI-2011-1071), Comunidad de Madrid (grant QUITEMAD+-CM, ref. S2013/ICE-2801) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 648913). This work was made possible through the support of grant no. 48322 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

Author information

Affiliations

  1. Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK

    • Toby S. Cubitt
  2. DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK

    • Toby S. Cubitt
  3. Departamento de Análisis Matemático and IMI, Facultad de CC Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain

    • David Perez-Garcia
  4. ICMAT, C/Nicolás Cabrera, Campus de Cantoblanco, 28049 Madrid, Spain

    • David Perez-Garcia
  5. Department of Mathematics, Technische Universität München, 85748 Garching, Germany

    • Michael M. Wolf

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Contributions

All authors contributed extensively to the paper.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Toby S. Cubitt.

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    Supplementary Information

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https://doi.org/10.1038/nature16059

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