### Abstract

Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling of the Hamiltonian gap at quantum first-order transitions, generally expected to be exponentially small in the size of the system. However, we show that a quantum antiferromagnetic Ising chain in a staggered field can exhibit a first-order transition with only an algebraically small gap. In addition, we construct a simple classical translationally invariant one-dimensional Hamiltonian containing nearest-neighbor interactions only, which exhibits an exponential gap at a thermodynamic quantum first-order transition of essentially topological origin. This establishes that (i) the QAA can be successful even across first-order transitions but also that (ii)it can fail on exceedingly simple problems readily solved by inspection, or by classical annealing.

Original language | English (US) |
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Article number | 030502 |

Journal | Physical review letters |

Volume | 109 |

Issue number | 3 |

DOIs | |

State | Published - Jul 19 2012 |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

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## Cite this

*Physical review letters*,

*109*(3), [030502]. https://doi.org/10.1103/PhysRevLett.109.030502