  Exact calculation and even multiplicative error estimation of matrix
permanent are challenging for both classical and quantum computers. Regarding
the permanents of random Gaussian matrices, the additive error estimation is
closely linked to boson sampling, and achieving multiplicative error estimation
requires exponentially many samplings. Our newly developed formula for matrix
permanents and its corresponding quantum expression have enabled better
estimation of the average additive error for random Gaussian matrices compared
to Gurvits' classical sampling algorithm. The well-known Ryser formula has been
converted into a quantum permanent estimator. When dealing with real random
Gaussian square matrices of size $N$, the quantum estimator can approximate the
matrix permanent with an additive error smaller than
$\epsilon(\sqrt{\mathrm{e}N})^{N}$, where $\epsilon$ is the estimation
precision. In contrast, Gurvits' classical sampling algorithm has an estimation
error of $\epsilon(2\sqrt{N})^{N}$, which is exponentially larger ($1.2^{N}$)
than the quantum method. As expected, the quantum additive error bound fails to
reach the multiplicative error bound of $(2\pi
N)^{1/4}\epsilon(\sqrt{N/\mathrm{e}})^{N}$. Additionally, the quantum permanent
estimator can be up to quadratically faster than the classical estimator when
using quantum phase estimation-based amplitude estimation.
