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

TL;DR

The paper introduces a factorization identity for twisted multinomial coefficients under certain conditions, extending classical formulas and applying it to quantum interferometry to produce exact matrix product states.

Contribution

It presents a new combinatorial identity for twisted multinomial coefficients with multiple parameters, generalizing existing formulas and connecting to quantum state preparation.

Findings

The identity factorizes twisted multinomials into products of Gaussian binomials with site-dependent parameters.

It applies to quantum algorithms for preparing Gibbs and ground states in Hamiltonian Decoded Quantum Interferometry.

The factorization yields exact matrix product states of bounded bond dimension for polynomial expansions.

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 $\omega_{ij}$ 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.