Fast mixing of all-to-all quantum systems at high temperatures
summary
Proves that bounded-degree all-to-all k‑local Hamiltonians have a critical temperature above which the Gibbs sampler has a system-size independent spectral gap, and that 1D spin chains have such a gap at all finite temperatures. Key tools are Gaussian‑filtered CKG updates, cluster expansions and a novel Dirichlet‑form comparison using a non‑CPTP pseudo‑Lindbladian K with an explicit local kernel; this yields a quantum FPTAS for partition functions and, in 1D, a polylog n depth circuit for Gibbs states and quasi‑polynomial classical simulation, but K is not physically implementable and beta‑dependence can be very bad.
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chapters (12)
- 0:00Fast mixing in quantum systems
- 1:26All-to-all quantum systems setup
- 8:14Spectral gaps and applications
- 15:32Prior spectral gap work
- 18:26Markov chain structure
- 27:04Dirichlet forms and gaps
- 33:09Local fixed points challenges
- 40:05New mixing time framework
- 44:12Proof outline and comparison
- 47:47Quantum/classical simulation
- 54:28Open problems
- 57:22Summary
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