The quantum adiabatic algorithm and scaling of gaps at first order quantum phase transitions

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-neighbour 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.

6 pages, 3 figures

References in corpus (5)

• Size dependence of the minimum excitation gap in the Quantum Adiabatic Algorithm
• Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve
• Simple Glass Models and their Quantum Annealing
• Cavity method for quantum spin glasses on the Bethe lattice
• A solvable model of quantum random optimization problems

Cited by in corpus (1)

• The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective