[seqfan] BosonSampling and Pi^2/6

Charles Greathouse charles.greathouse at case.edu
Wed Aug 21 08:22:43 CEST 2013

Scott Aaronson wrote a pseudo-tweet on his blog last week:
While working on BosonSampling yesterday, (1/6)pi^2 and Euler-Mascheroni
constant made unexpected unappearances. What I live for

He expanded in the comments:
If you care, π^2/6 and γ both arose from calculating the moments of
ln(X_n), where X_n is a chi-squared random variable with n degrees of
freedom (or actually, its complex generalization -- i.e., a sum of absolute
squares of n independent complex Gaussians with mean 0 and variance 1).
Calculating those moments let me understand the distribution of the product
of squared 2-norms of the rows of an iid Gaussian matrix. And that, in
turn, let me prove that the BosonSampling distribution can be distinguished
from the uniform distribution in classical polynomial time.

I don't quite follow -- maybe because it's 2:30 AM. Is there anything that
should be added to A001620 and/or A013661?

Charles Greathouse
Case Western Reserve University

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