The key challenge in efficient quantum Gibbs sampling is slow mixing: many systems associated with a phase transition are known to have exponential mixing time at low temperature, which raises the question of whether quantum Gibbs sampling algorithms can still do anything interesting for these systems. However, mixing time is fundamentally a worst-case notion and does not rule out fast convergence to the Gibbs state when starting from a good initial state. We argue that the simple idea of quench dynamics (running the Gibbs sampler starting from the maximally mixed state) can bypass slow-mixing for the 4D toric code and converge to the low-temperature Gibbs state in polynomial time. From the physics perspective, this suggests that nature can prepare a topological qubit by itself, within a thermal noise environment.
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