Complexity of counting products of commuting operators drawn from a specific set. I have run into the following problem in the course of trying to solve another problem, and have been unable to come up with a decent answer. This may reduce trivially to a known problem, but if so I am not seeing it, and would be extremely thankful if someone could point me in the right direction.

Given a set of unique $n \times n$ matrices $\{\lambda^{i}\}$ of the form $\lambda^{i}_{jk} = \delta_{jk} (1-2\delta_{x_i j}-2\delta_{y_i j})$, where $1\leq x_i &lt; y_i \leq n$, is there an efficient way to calculate $N_D$, the number of operators of the form $C_b = \prod_i \left(\lambda^i \right)^{b_i}$ which satisfy $\mbox{Tr}(C_b) = D$? Here $b_i \in \{0,1\}$. I would be interested to know if there is any polynomial time algorithm, or even if it is in P/poly, or alternatively, if this reduces to some more studied open problem.

As things stand, I can currently obtain $\sum_D N_D$ and $\sum_D N_D D$, but not enough other linearly independent equations to be able to solve individually for each $N_D$.
For $D$ unrestricted, computing $N_D$ exactly is equivalent to evaluating the Tutte polynomial of a graph: the values $N_D$ determine the weight enumerator of a linear code, also known as the Tutte polynomial of a binary matroid, and your restriction on the $\lambda^i$ corresponds to the linear code being given by a basis of vectors of Hamming weight 2, which is the just the usual cycle matroid of a graph.

On the other hand, we can calculate $N_{n-2k}$ if $k$ is constant in polynomial time as follows. There are $O(n^k)$ diagonal matrices $C$ with $n-k$ ones and $k$ minus-ones. These can be enumerated in polynomial time for fixed $k$. $C$ can be considered as a point in $\mathbb{F}_2^n$ and each $b$ can be considered as a point in $\mathbb{F}_2^m$ where $m$ is the number of $\lambda^i$s. This turns multiplication into addition, so the definition of $C_b$ becomes $C_{b_j}=\sum \lambda^i_{jj} b_i$. In Diestel's book on graph theory the values $b$ such that $\sum \lambda^i_{jj} b_i=0$ are called the cycle space of the graph.