Preparing ground states or thermal states poses a significant challenge in quantum simulation algorithms. In this talk, we take a Markov Chain Monte Carlo (MCMC) approach to sample quantum Gibbs states. We introduce the first construction of a continuous-time quantum Markov chain (i.e., Lindbladian) which simultaneously satisfies the following properties: exact quantum detailed balance; efficient Lindbladian simulation at cost linear in the evolution time and the inverse temperature and polylogarithmically with the precision; purification as a ``quantum-walk'' Hamiltonian whose ground state is the purified Gibbs state. Furthermore, for lattice Hamiltonians, our Lindbladian decomposes into a sum of (quasi)-local Lindblad operators whose radius scales with the inverse temperature, giving favorable practical gate complexities. The combination of nice features suggests our construction is the ideal quantum counterpart of classical MCMC. This is based on joint work with Michael Kastoryano, Fernando Brandao, and Andras Gilyen and aims to provide a perspective that complements open system thermodynamics.
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