Making both ends meet: from efficient simulation to universal quantum computing with quantum Gibbs sampling
Résumé
The preparation of thermal states of matter is a crucial task in quantum simulation. In this work, we prove that an efficiently implementable dissipative evolution recently introduced by Chen et al. thermalizes into its equilibrium Gibbs state in time scaling polynomially with system size at high enough temperatures for any Hamiltonian that satisfies a Lieb-Robinson bound, such as local Hamiltonians on a lattice. Furthermore, we show the efficient adiabatic preparation of the associated purifications or ``thermofield double" states. To the best of our knowledge, these are the first results rigorously establishing the efficient preparation of high temperature Gibbs states and their purifications. In the low-temperature regime, we show that implementing this family of Lindbladians for inverse temperatures logarithmic in the system's size is polynomially equivalent to standard quantum computation. On a technical level, for high temperatures, our proof makes use of the mapping of the generator of the evolution into a Hamiltonian and the analysis of the stability of its gap. For low temperature, we instead perform a perturbation at zero temperature of the Laplace transform of the energy observable at fixed runtime, and resort to circuit-to-Hamiltonian mappings akin to the proof of universality of quantum adiabatic computing. Taken together, our results show that the family of Lindbladians of Chen et al. efficiently prepares a large class of quantum many-body states of interest, and have the potential to mirror the success of classical Monte Carlo methods for quantum many-body systems.