Quantum adiabatic computation with a constant gap is not useful in one dimension

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We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computation. Therefore, adiabatic algorithms which are ... continued below

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050502

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Hastings, Matthew January 1, 2009.

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We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computation. Therefore, adiabatic algorithms which are useful for universal quantum computation either require a spectral gap tending to zero or need to be implemented in more than one dimension (we leave open the question of the computational power of adiabatic simulation with a constant gap in more than one dimension).

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050502

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  • Journal Name: Physical Review Letters; Journal Volume: 103; Journal Issue: 5

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  • Report No.: LA-UR-09-00691
  • Report No.: LA-UR-09-691
  • Grant Number: AC52-06NA25396
  • DOI: 10.1103/PhysRevLett.103.050502 | External Link
  • Office of Scientific & Technical Information Report Number: 956377
  • Archival Resource Key: ark:/67531/metadc934221

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  • January 1, 2009

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  • Nov. 13, 2016, 7:26 p.m.

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  • Dec. 12, 2016, 12:48 p.m.

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Hastings, Matthew. Quantum adiabatic computation with a constant gap is not useful in one dimension, article, January 1, 2009; [New Mexico]. (digital.library.unt.edu/ark:/67531/metadc934221/: accessed October 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.