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 $\Omega$. In general, no closed-form evaluation is known. Our main result is that under a natural structural condition on $\Omega$ - 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: $\binom{k}{k_1,\ldots,k_m}_\Omega = \prod_j\binom{\ell_j}{k_j}_{q_j}$ where $\ell_j = k_1+\cdots+k_j$. 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 $\mathrm{deg}(\mathcal{P})+1$ for the expansion coefficients of $\mathcal{P}(h)$, for any polynomial $\mathcal{P}$, via a polynomial-dependent right boundary vector.