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We study both classical and quantum algorithms to solve a hard optimizationproblem, namely 3-XORSAT on 3-regular random graphs. By introducing a newquasi-greedy algorithm that is not allowed to jump over large energy barriers,we show that the problem hardness is mainly due to entropic barriers. We study,both analytically and numerically, several optimization algorithms, findingthat entropic barriers affect in a similar way classical local algorithms andquantum annealing. For the adiabatic algorithm, the difficulty we identify isdistinct from that of tunnelling under large barriers, but does, nonetheless,give rise to exponential running (annealing) times.

Entropic barriers as a reason for hardness in both classical and quantum algorithms