Fine-grained quantum supremacy of the one-clean-qubit model
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The one-clean-qubit model (or the DQC1 model) is a restricted model of quantum computing where all but a single input qubits are maximally mixed. It is known that output probability distributions of the DQC1 model cannot be classically sampled in polynomial-time unless the polynomial-time hierarchy collapses. In this paper, we show that even superpolynomial-time and exponential-time classical samplings are impossible under certain fine-grained complexity conjectures. We also show similar fine-grained quantum supremacy results for the Hadamard-classical circuit with one-qubit (HC1Q) model, which is another sub-universal model with a classical circuit sandwiched by two Hadamard layers.
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