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This paper shows that many quantum systems where every particle can interact with many others — so-called all-to-all k-local systems — settle to thermal equilibrium at a rate that does not slow down as the system grows, provided the temperature is high enough. In plain terms, the authors prove that a standard model of thermalization in open quantum systems has a spectral gap that stays bounded away from zero even as the number of particles increases. A spectral gap here measures how fast the system forgets its initial state and approaches the thermal or Gibbs state.
The authors study n quantum d-level particles (qudits) with terms in the energy operator (Hamiltonian) that each act on at most k particles. They assume the interaction degree (how many partners each particle couples to) and a suitable measure of interaction strength are bounded so these quantities do not grow with n. They analyze a specific continuous-time thermalization model (a Lindbladian generator introduced in earlier work) and prove that this generator has a system-size independent gap whenever the inverse temperature β is below a threshold β_c. Concretely, the threshold scales like β_c = [c · q · k · d · J]^{-1} for a universal constant c, where q is the local dimension, k the locality, d the degree, and J the interaction strength. At high temperatures (small β) the bound applies.
Why this matters: a size-independent spectral gap implies efficient ways to approximate thermal quantities on a quantum computer. The paper shows that, under the same high-temperature condition, one can estimate the partition function (a normalizing constant that encodes thermodynamic information) and global expectation values to fixed accuracy in time polynomial in n and 1/ε. In the authors’ analysis the time to get within ε in trace distance satisfies t_mix(ε) = O(n + log(1/ε)), so the result yields an efficient state-preparation route by simulating the Lindbladian dynamics. For the common case of two-level systems with pairwise interactions, the threshold β_c becomes a universal constant; the authors note this is believed to be essentially optimal given known hardness results for related counting problems.
Important caveats and limits: the result only holds at sufficiently high temperature (β smaller than the stated threshold). The proof guarantees a spectral gap and so “fast mixing,” but it does not prove the stronger property known as “rapid mixing,” which would give mixing times logarithmic in system size. The argument relies on a particular Lindbladian construction and on assumptions that k, the local dimension q, and the product d·J remain bounded independent of n. The methods used are not aimed at disordered or glassy models; the paper notes that much more work would be needed to handle random spin glasses or related strongly correlated all-to-all models.
On technique and outlook: to get these statements the authors combine a quantum version of a cluster expansion (a way of controlling the short-time complex evolution of operators) with non-commutative forms of Dobrushin conditions (sufficient criteria for mixing). They adapt earlier ideas used for lattice systems and show they extend to non-geometric, long-range interaction settings. The paper points to possible follow-up uses of these tools, including improved classical algorithms for some models, but it does not attempt those applications here.