It is well known that Ryser's algorithm requires O(Nx2^N) operations for computation of a matrix permanent of a square complex matrix of size NxN on an *ideal computer with exact arithmetics*. However, I am interested in practical computation, that means on a real computer with, say,  IEEE arithmetics. One should take into account the roundoff and truncation errors. To make the problem   precise: Consider random unitary matrix of size NxN and a relative error epsilon, give estimate on the number of operations for computation of per(U) to the error   epsilon*|per(U)|  [for nonzero per(U)].  

Say we have a 64-bit standard IEEE machine. How the number of flops scales with N  and the  inverse relative error 1/epsilon.  [I guess, a probabilistic answer, i.e., to a probability 1- delta we get the result in the above form, is only possible. Then what is delta(N,epsilon)?] 

P.S. 

I am actually interested not in random unitary matrices, but in random NxN matrices built from  a unitary, say nxn, n fixed,  one taking with repetitions its rows/columns. There are estimates on the number of flops in the exact arithmetics in 

https://www.researchgate.net/publication/236274558_Asymptotic_evaluation_of_bosonic_probability_amplitudes_in_linear_unitary_networks_in_the_case_of_large_number_of_bosons (see the Appendix).

Such matrices appear also in Bayesian Quantum Tomography. http://arxiv.org/abs/1403.5158

But the problem is similar and simpler for random unitaries. 

Article Asymptotic evaluation of bosonic probability amplitudes in l...