Quantum Gibbs State Preparation via Relative Decoding: From Fast-Mixing Sources to Broader Target Classes

Provable guarantees for preparing quantum Gibbs states, such as rapid mixing or gapped coherent preparation paths, are known only for restricted Hamiltonian families and usually must be re-derived for each new family. Building on homomorphic polynomial transduction, a generalization of decoded quantum interferometry (DQI) and Hamiltonian DQI (HDQI), we show how relative decoding transfers such a guarantee from a source Hamiltonian $H_A$ to a target $H_B$. Writing each Hamiltonian as a sum of $m$ Pauli terms, the subsets of terms whose product is proportional to the identity, its relations, form a binary linear code, $K_A$ for the source and $K_B$ for the target, with the Pauli-label matrix as parity-check matrix as in DQI. Starting from the canonical thermofield double (TFD) of $H_A$, a Bell transform, a reversible label map, and a coherent decoder for the quotient code $K_B\mathrm{/}{K}_A$ prepare an approximate TFD of $H_B$, and hence its Gibbs state. Because this decoder only resolves target relations absent from the source, the reachable inverse temperature is set by the relative distance $d_{\mathrm{r}\mathrm{e}\mathrm{l}}$, the fewest terms in any such relation. It can far exceed the ordinary distance $d_{\mathrm{o}\mathrm{r}\mathrm{d}}$, the fewest terms in any target relation, which bounds the uniform exact decoding radius of DQI and HDQI. We show that linear $d_{\mathrm{r}\mathrm{e}\mathrm{l}}$ with efficient decoding at linear radius certifies a constant inverse temperature. As an example, starting from a nearest-neighbor spin chain whose TFD is known to be preparable at every finite temperature, a sparse classical parity-check matrix yields bounded-degree, geometrically nonlocal, noncommuting targets with $d_{\mathrm{o}\mathrm{r}\mathrm{d}}=3$ and $d_{\mathrm{r}\mathrm{e}\mathrm{l}}=\mathrm{\Theta }(m)$. To our knowledge, this gives a new class of efficiently preparable TFDs.