Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy
share
It is a long-standing open problem whether quantum computing can be verified by a classical verifier. In the computational complexity term, it is "Does any BQP problem have an interactive proof system with a BQP prover and a BPP verifier?". Several partial solutions to the open problem have been obtained. For example, verifiable blind quantum computing protocols demonstrate that if the verifier is slightly quantum, the problem is solved. Furthermore, the answer to the open problem is yes for specific BQP problems, such as the recursive Fourier sampling. In this paper, we consider a problem of distinguishing output probability distributions of two quantum circuits. We show that the problem is BQP-complete, but if the two circuits are restricted to some circuits in the second level of the Fourier hierarchy, such as IQP circuits, the problem has a Merlin-Arthur proof system where the prover (Merlin) has the power of BQP for yes instances. Our result is therefore another example of specific BQP problems that have an interactive proof system with a BQP prover and a BPP verifier. As an additional result, we also consider a simple model in the second level of the Fourier hierarchy, and show that a multiplicative-error classical efficient sampling of the output probability distribution of the model causes the collapse of the polynomial-time hierarchy to the third or the second level. The model consists of a classical reversible circuit sandwiched by two layers of Hadamard gates. The proof technique for its quantum supremacy is different from those for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for other applications.
READ FULL TEXT