Hamiltonian Decoded Quantum Interferometry for General Pauli Hamiltonians
Kaifeng Bu, Weichen Gu, Xiang Li
TL;DR
This work develops Hamiltonian Decoded Quantum Interferometry (HDQI) for general Pauli Hamiltonians $H=\sum_i c_i P_i$, showing that, with a suitable decoding oracle, one can efficiently prepare the density $\rho_{\mathcal P}(H)=\mathcal P^2(H)/\mathrm{Tr}[\mathcal P^2(H)]$ for any univariate polynomial $\mathcal P$, enabling Gibbs-state-like sampling when $\mathcal P$ approximates $e^{-\beta x/2}$. The authors unify commuting, nearly independent commuting, and noncommuting Pauli terms under a matrix-product-state (MPS) reference-state framework, providing explicit constructions: $|R^l(H)\rangle$ for the commuting case with bond $D=l+1$, $|R^l_k(H)\rangle$ for dimension-$k$ symplectic codes with $D=2^k(l+1)$, and $|R^l_*(H)\rangle$ for noncommuting terms with $D=l+1$ and width $q=2^{\mathcal M}$. Complexity bounds show polynomial-time pre-processing and state-preparation after a single decoding call, with robustness guarantees against imperfect decoding that scale as $O(\sqrt{\epsilon})$ in trace distance. The results broaden HDQI beyond stabilizer-like Hamiltonians and suggest practical pathways for Gibbs-state preparation and Hamiltonian optimization in broad quantum systems. The work also highlights open questions on semicircular laws for general Pauli Hamiltonians and potential ground-state applications.
Abstract
In this work, we study the Hamiltonian Decoded Quantum Interferometry (HDQI) for the general Hamiltonians $H=\sum_ic_iP_i$ on an $n$-qubit system, where the coefficients $c_i\in \mathbb{R}$ 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 $ρ_{\mathcal P}(H) = \frac{\mathcal P^2(H)}{\text{Tr}[\mathcal P^2(H)]}$, 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.