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Quantum annealing (QA) is one of the methods to solve combinatorialoptimization problems. We can estimate a computational time of QA by using theso-called adiabatic condition derived from the adiabatic theorem. The adiabaticcondition consists of two parts: an energy gap and a transition matrix. It isbelieved that the computational time mainly depends on the energy gap during QAand is inversely proportional to a polynomial of the minimal energy gap. Inthis paper, we challenge this common wisdom. We propose a general method toconstruct counterintuitive models with a constant energy gap during QA where QAwith a constant annealing time fails despite a constant energy gap. In ourformalism, we choose a known model exhibiting an exponentially small energy gapduring QA, and we modify the model by adding a specific penalty term to theHamiltonian. In the modified model, the transition matrix in the adiabaticcondition becomes exponentially large with the number of qubits, while theenergy gap remains constant. As concrete examples we consider the adiabaticGrover search and the ferromagnetic p-spin model. In these cases, by adding thepenalty term, the success probability of QA in the modified models becomeexponentially small despite a constant energy gap. Our results paves a way forbetter understanding of the QA performance.

Quantum annealing showing the exponentially small success probability despite a constant energy gap