  The remainder $E_\Gamma(X)$ in the Prime Geodesic Theorem for the Picard
group $\Gamma = \mathrm{PSL}(2,\mathbb{Z}[i])$ is known to be bounded by
$O(X^{3/2+\epsilon})$ under the assumption of the Lindel\"of hypothesis for
quadratic Dirichlet $L$-functions over Gaussian integers. By studying the
second moment of $E_\Gamma(X)$, we show that on average the same bound holds
unconditionally.
