Gaussian Noise Sensitivity and BosonSampling

We study the sensitivity to noise of |permanent(X)|2 for random real and complex n×n Gaussian matrices X, and show that asymptotically the correlation between the noisy and noiseless outcomes tends to zero when the noise level is ω(1)/n. This suggests that, under certain reasonable noise models, the probability distributions produced by noisy BosonSampling are very sensitive to noise. We also show that when the amount of noise is constant the noisy value of |permanent(X)|2 can be approximated efficiently on a classical computer. These results seem to weaken the possibility of demonstrating quantum-speedup via BosonSampling without quantum fault-tolerance. ∗This work was carried out in part while the authors were visiting the Simons Institute for the Theory of Computing at UC Berkeley. †Einstein Institute of Mathematics, the Hebrew University of Jerusalem, and Department of Mathematics, Yale University. Supported by an ERC grant and by an NSF grant. ‡School of Computer Science and Engineering, the Hebrew University. Supported by an Israeli Science Fund grant and a Binational Science Fund grant no. 2008477.