# Sample complexity of device-independently certified "quantum supremacy"

## TL;DR

This paper investigates the difficulty of certifying quantum sampling experiments, showing that non-interactive classical certification requires exponentially many samples due to the distributions' properties, impacting claims of quantum supremacy.

## Contribution

It proves that non-interactive certification of quantum sampling distributions demands exponential resources, highlighting fundamental limitations for certifying quantum supremacy experiments.

## Key findings

- Non-interactive certification requires exponential samples.
- Sampling distributions have small second moments.
- Certification difficulty applies to boson sampling, IQP, and random circuits.

## Abstract

Results on the hardness of approximate sampling are seen as important stepping stones towards a convincing demonstration of the superior computational power of quantum devices. The most prominent suggestions for such experiments include boson sampling, IQP circuit sampling, and universal random circuit sampling. A key challenge for any such demonstration is to certify the correct implementation. For all these examples, and in fact for all sufficiently flat distributions, we show that any non-interactive certification from classical samples and a description of the target distribution requires exponentially many uses of the device. Our proofs rely on the same property that is a central ingredient for the approximate hardness results: namely, that the sampling distributions, as random variables depending on the random unitaries defining the problem instances, have small second moments.