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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.

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  • Ground State Energy
  • Quantum Circuit
  • Full Version
  • Local Unitaries
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Correspondence to Jamie Sikora .

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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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