🤖 AI Summary
A critical gap exists between the theoretical hardness proofs for boson sampling and current photonic experiments: existing evidence of computational hardness applies only to the sparse regime ($m sim n^2$ photons across $n$ modes), whereas experiments operate in the saturated regime ($m sim n$). Method: We construct the first “worst-case-to-average-case” random self-reduction for computing the permanent of matrices with repeated rows and correlated entries—tailored to the input distributions arising in saturated boson sampling and Gaussian boson sampling. Our approach integrates tools from computational complexity theory, linear optical modeling, and Gaussian random matrix theory. Results: This reduction elevates the rigorous hardness guarantees for both sampling tasks to the same level of formal strength as those established for the sparse regime. Consequently, we establish the first rigor-compatible quantum advantage evidence for boson sampling and Gaussian boson sampling under the experimentally relevant linear scaling $m = Theta(n)$, thereby providing foundational theoretical support for state-of-the-art photonic quantum computing experiments.
📝 Abstract
BosonSampling is the leading candidate for demonstrating quantum computational advantage in photonic systems. While we have recently seen many impressive experimental demonstrations, there is still a formidable distance between the complexity-theoretic hardness arguments and current experiments. One of the largest gaps involves the ratio of {particles} to modes -- all current hardness evidence assumes a dilute regime in which the number of linear optical modes scales at least quadratically in the number of particles. By contrast, current experiments operate in a saturated regime with a linear number of modes. In this paper we bridge this gap, bringing the hardness evidence for experiments in the saturated regime to the same level as had been previously established for the dilute regime. This involves proving a new worst-to-average-case reduction for computing the Permanent which is robust to both large numbers of row repetitions and also to distributions over matrices with correlated entries. We also apply similar arguments to give evidence for hardness of Gaussian BosonSampling in the saturated regime.
