Computing the determinant of a matrix can be done in polynomial time, while computing the permanent is known to be #P-hard. Let $A$ be an $n \times n$ matrix. Define a generalised determinant function as follows,
$$ Det_ \theta (A) = \sum_ {\mu \in S_n} e^{i \theta ~ sign(\mu)} A_{1,\mu (1)} \ldots A_ {n , \mu(n)}$$.
$S_n$ is the symmetric group on $n$ elements. It can be readily seen that $Det_0 (A) = Perm(A)$ and $Det_{\pi/2}(A) = i Det(A)$.
- For what value of $\theta$ does the computation of $Det_\theta{A}$ cease to be #P-hard?
- For what value of $\theta$ does $Det_\theta{A}$ become poly time computable?
- As $\theta$ varies how does the complexity of computation of $Det_\theta{A}$ change in general?
ps - Some physics- Prima facie $Det_\theta{A}$ looks like the analogue of Slater determinant corresponding to Anyon wavefunctions.