A Factorization Identity for Twisted Multinomial Coefficients with Application to Pilot States in Hamiltonian Decoded Quantum Interferometry

The $q$-multinomial coefficient, a classical object in enumerative combinatorics, counts permutations of multisets weighted by the number of inversions, with a single deformation parameter $q$. We introduce the twisted multinomial coefficient, in which each inversion between letters $i$ and $j$ carries a pair-dependent weight ${\omega }_{ij}$ determined by a skew-symmetric matrix $\mathrm{\Omega }$. In general, no closed-form evaluation is known. Our main result is that under a natural structural condition on $\mathrm{\Omega }$ - predecessor-uniformity (${\omega }_{ij}=q_j$ for all $i<j$) - the twisted multinomial factorizes as a product of Gaussian ($q$-deformed) binomials with site-dependent parameters: ${\left(\genfrac{}{}{0px}{}{k}{k_1,\dots ,k_m}\right)}_{\mathrm{\Omega }}={\prod }_j{\left(\genfrac{}{}{0px}{}{{\mathrm{\ell }}_j}{k_j}\right)}_{q_j}$ where ${\mathrm{\ell }}_j=k_1+\cdots +k_j$. This extends the standard product formula for the $q$-multinomial from a single parameter $q$ to $m-1$ independent parameters. The identity is purely combinatorial: it holds for arbitrary $q_j\in \mathbb{C}\setminus \{0\}$ without any algebraic constraints. We were led to this identity by studying pilot state preparation in Hamiltonian Decoded Quantum Interferometry (HDQI), a recently proposed quantum algorithm for preparing Gibbs and ground states. As an application, we show that the factorization yields an exact matrix product state (MPS) of bond dimension $k+1$ for the expansion coefficients of $h^k$ in a twisted algebra. We further show that the same site matrices deliver an exact MPS of bond dimension $\mathrm{d}\mathrm{e}\mathrm{g}(\mathcal{P})+1$ for the expansion coefficients of $\mathcal{P}(h)$, for any polynomial $\mathcal{P}$, via a polynomial-dependent right boundary vector.