Fine-grained quantum supremacy based on Orthogonal Vectors, 3-SUM and All-Pairs Shortest Paths.

Fine-grained quantum supremacy is a study of proving (nearly) tight time lower bounds for classical simulations of quantum computing under "fine-grained complexity" assumptions. We show that under conjectures on Orthogonal Vectors (OV), 3-SUM, All-Pairs Shortest Paths (APSP) and their variants, strong and weak classical simulations of quantum computing are impossible in certain exponential time with respect to the number of qubits. Those conjectures are widely used in classical fine-grained complexity theory in which polynomial time hardness is conjectured. All previous results of fine-grained quantum supremacy are based on ETH, SETH, or their variants that are conjectures for SAT in which exponential time hardness is conjectured. We show that there exist quantum circuits which cannot be classically simulated in certain exponential time with respect to the number of qubits first by considering a Quantum Random Access Memory (QRAM) based quantum computing model and next by considering a non-QRAM model quantum computation. In the case of the QRAM model, the size of quantum circuits is linear with respect to the number of qubits and in the case of the non-QRAM model, the size of the quantum circuits is exponential with respect to the number of qubits but the results are still non-trivial.

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