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International Colloquium on Automata, Languages, and Programming

ICALP 2015: Automata, Languages, and Programming pp 617–628Cite as

Ground State Connectivity of Local Hamiltonians

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9134)

Abstract

The study of ground state energies of local Hamiltonians has played a fundamental role in quantum complexity theory. In this paper, we take a new direction by introducing the physically motivated notion of “ground state connectivity” of local Hamiltonians, which captures problems in areas ranging from quantum stabilizer codes to quantum memories. We show that determining how “connected” the ground space of a local Hamiltonian is can range from QCMA-complete to PSPACE-complete, as well as NEXP-complete for an appropriately defined “succinct” version of the problem. As a result, we obtain a natural QCMA-complete problem, a goal which has generally proven difficult since the conception of QCMA over a decade ago. Our proofs rely on a new technical tool, the Traversal Lemma, which analyzes the Hilbert space a local unitary evolution must traverse under certain conditions. We show that this lemma is essentially tight with respect to the length of the unitary evolution in question.

Keywords

  • Ground State Energy
  • Quantum Circuit
  • Full Version
  • Local Unitaries
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Authors and Affiliations

  1. Department of Computer Science, Virginia Commonwealth University, Richmond, USA

    Sevag Gharibian

  2. Centre for Quantum Technologies and MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, National University of Singapore, Singapore, Singapore

    Jamie Sikora

Authors
  1. Sevag Gharibian
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  2. Jamie Sikora
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Corresponding author

Correspondence to Jamie Sikora .

Editor information

Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Gharibian, S., Sikora, J. (2015). Ground State Connectivity of Local Hamiltonians. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_50

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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_50

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