Hamiltonian Decoded Quantum Interferometry for General Pauli Hamiltonians

In this work, we study the Hamiltonian Decoded Quantum Interferometry (HDQI) for the general Hamiltonians H=sum_iciPi on an n-qubit system, where the coefficients c_iin mathbb{R} and P_i are Pauli operators. We show that, given access to an appropriate decoding oracle, there exist efficient quantum algorithms for preparing the state ρ_{mathcal P}(H) = frac{mathcal P²(H)}{Tr\[mathcal P²(H)\]}, where mathcal P(H) denotes the matrix function induced by a univariate polynomial mathcal P(x). Such states can be used to approximate the Gibbs states of H for suitable choices of polynomials. We further demonstrate that the proposed algorithms are robust to imperfections in the decoding procedure. Our results substantially extend the scope of HDQI beyond stabilizer-like Hamiltonians, providing a method for Gibbs-state preparation and Hamiltonian optimization in a broad class of physically and computationally relevant quantum systems.