A Factorization Identity for Twisted Multinomial Coefficients with Application to Pilot States in Hamiltonian Decoded Quantum Interferometry
Pawel Wocjan
Abstract
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 $ω_{ij}$ determined by a skew-symmetric matrix $Ω$. In general, no closed-form evaluation is known. Our main result is that under a natural structural condition on $Ω$ - predecessor-uniformity ($ω_{ij} = q_j$ for all $i<j$) - the twisted multinomial factorizes as a product of Gaussian ($q$-deformed) binomials with site-dependent parameters: $\binom{k}{k_1,\ldots,k_m}_Ω= \prod_j\binom{\ell_j}{k_j}_{q_j}$. 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 note that this addresses only one component of the HDQI pipeline (pilot state preparation); the full protocol additionally requires efficient decoding of the associated Hamiltonian code, and both components must work in conjunction for Hamiltonians of physical interest. Identifying such Hamiltonians remains an important open problem.