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Undecidability of the spectral gap.

Toby S Cubitt1,2, David Perez-Garcia3,4, Michael M Wolf5

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

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The spectral gap problem in quantum many-body physics is undecidable. This means no general algorithm can determine if a quantum system is gapped or gapless, impacting fundamental physics research.

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Area of Science:

  • Quantum Many-Body Physics
  • Computational Complexity Theory

Background:

  • The spectral gap, the energy difference between ground and excited states, is crucial for understanding quantum systems.
  • Many significant problems in physics, including the Haldane conjecture and topological phases, revolve around spectral gaps.

Purpose of the Study:

  • To determine the decidability of the spectral gap problem for quantum many-body systems.
  • To investigate the implications of undecidability for other low-energy properties.

Main Methods:

  • Construction of quantum spin systems on a 2D lattice with specific interaction types.
  • Utilizing Hamiltonian complexity and aperiodic tilings.
  • Encoding a quantum phase-estimation algorithm and a universal Turing machine within the system's ground state.

Main Results:

  • The spectral gap problem for quantum spin systems with translationally invariant, nearest-neighbor interactions is proven to be undecidable.
  • This undecidability extends to other low-energy properties like ground-state correlation decay.
  • The spectral gap's determination is linked to the halting problem's outcome.

Conclusions:

  • There is no universal algorithm to ascertain if an arbitrary quantum model is gapped or gapless.
  • Certain quantum models exhibit spectral gap properties independent of mathematical axioms.