  We generalise a result of Bykovskii to the Gaussian integers and prove an
asymptotic formula for the prime geodesic theorem in short intervals on the
Picard manifold. Previous works show that individually the remainder is bounded
by $O(X^{13/8+\epsilon})$ and $O(X^{3/2+\theta+\epsilon})$, where $\theta$ is
the subconvexity exponent for quadratic Dirichlet $L$-functions over
$\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral
exponential sum and a smooth explicit formula, we obtain an improvement for
both of these exponents. Moreover, by assuming two standard conjectures on
$L$-functions, we show that it is possible to reduce the exponent below the
barrier $3/2$ and get $O(X^{34/23+\epsilon})$ conditionally. We also
demonstrate a dependence of the remainder in the short interval estimate on the
classical Gauss circle problem for shifted centres.
