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We introduce a simple sub-universal quantum computing model, which we callthe Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of aclassical reversible circuit sandwiched by two layers of Hadamard gates, andtherefore it is in the second level of the Fourier hierarchy. We show thatoutput probability distributions of the HC1Q model cannot be classicallyefficiently sampled within a multiplicative error unless the polynomial-timehierarchy collapses to the second level. The proof technique is different fromthose used for previous sub-universal models, such as IQP, Boson Sampling, andDQC1, and therefore the technique itself might be useful for finding othersub-universal models that are hard to classically simulate. We also study theclassical verification of quantum computing in the second level of the Fourierhierarchy. To this end, we define a promise problem, which we call theprobability distribution distinguishability with maximum norm (PDD-Max). It isa promise problem to decide whether output probability distributions of twoquantum circuits are far apart or close. We show that PDD-Max is BQP-complete,but if the two circuits are restricted to some types in the second level of theFourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has aMerlin-Arthur system with quantum polynomial-time Merlin and classicalprobabilistic polynomial-time Arthur.

Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy