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We discuss the quantum annealing of the fully-connected ferromagnetic $ p$-spin model in a dissipative environment at low temperature. This model, inthe large $ p $ limit, encodes in its ground state the solution to the Grover'sproblem of searching in unsorted databases. In the framework of the quantumcircuit model, a quantum algorithm is known for this task, providing aquadratic speed-up with respect to its best classical counterpart. Thisimprovement is not recovered in adiabatic quantum computation for an isolatedquantum processor. We analyze the same problem in the presence of alow-temperature reservoir, using a Markovian quantum master equation inLindblad form, and we show that a thermal enhancement is achieved in thepresence of a zero temperature environment moderately coupled to the quantumannealer.

May a dissipative environment be beneficial for quantum annealing?