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Description
The field of quantum generative models based on Instantaneous Quantum Polynomial (IQP) circuits has seen significant interest in recent years, driven by results proving classical difficulty of sampling from sparse anti-concentrated IQP circuits and efficient trainability. Practical demonstrations of the former are crucial in establishing the superiority of these circuits in generating samples over classical methods. In this work, we show that Linear Cross Entropy benchmarking, the most widely considered test to demonstrate quantum supremacy, using IQP circuits can be spoofed efficiently and argue that beating the spoofer using quantum computers will require near fault-tolerant levels of fidelities, which puts such experiments firmly beyond the capabilities of current hardware. Furthermore, we introduce an algorithm that can compute amplitudes of large sparse IQP circuits consisting of 30-50 qubits in under 1 second on GPUs, which can be leveraged to generate samples in under a minute. As a final contribution, we show how one can generate peaked IQP circuits defined on hundreds of qubits efficiently, using which one can set up verifiable supremacy tasks, thus providing a better path towards the experimental demonstration of quantum supremacy using IQP circuits. Finally, we argue that such peaked circuits can be the cornerstone of classical difficulty arguments for IQP-based generative models, since we believe it is unlikely that flat anti-concentrated distributions, which are challenging to distinguish from uniform, will appear in practically relevant datasets.
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