Exponential improvements to the average-case hardness of BosonSampling

BosonSampling and Random Circuit Sampling are important both as a theoretical tool for separating quantum and classical computation, and as an experimental means of demonstrating quantum speedups. Prior works have shown that average-case hardness of sampling follows from certain unproven conjectures about the hardness of computing output probabilities, such as the Permanent-of-Gaussians Conjecture (PGC), which states that e-n log n-n-O(log n) additive-error estimates to the output probability of most random BosonSampling experiments are #P-hard. Prior works have only shown weaker average-case hardness results that do not imply sampling hardness. Proving these conjectures has become a central question in quantum complexity. In this work, we show that e-n log n-n-O(n^Δ) additive-error estimates to output probabilities of most random BosonSampling experiments are #P-hard for any Δ> 0, exponentially improving on prior work. In the process, we circumvent all known barrier results for proving PGC. The remaining hurdle to prove PGC is now 'merely' to show that the O(n^Δ) in the exponent can be improved to O(log n). We also obtain an analogous result for Random Circuit Sampling. We then show, for the first time, a hardness of average-case classical sampling result for BosonSampling, under an anticoncentration conjecture. Specifically, we prove the impossibility of multiplicative-error sampling from random BosonSampling experiments with probability 1-2-O(N^1/3) for input size N, unless the Polynomial Hierarchy collapses. This exponentially improves upon the state-of-the-art. To do this, we introduce new proof techniques which tolerate exponential loss in the worst-to-average-case reduction. This opens the possibility to show the hardness of average-case sampling without ever proving PGC.