Controlled efficient computation of high-temperature diffusion constants
In plain words
In a generic chain of interacting quantum spins at high temperature, energy and spin spread diffusively at a definite rate, but no method computes this rate with a guaranteed error at reasonable cost. Exact simulation becomes exponentially expensive because the quantum state grows highly entangled.
Precise statement
For a nonintegrable local spin chain (e.g. the mixed-field Ising chain $H=\sum_i J Z_i Z_{i+1}+h_x X_i+h_z Z_i$) at infinite temperature, compute the energy diffusion constant $D$ from the Kubo formula to relative accuracy $\epsilon$ at classical cost polynomial in $1/\epsilon$, with an a posteriori error estimate. Operator-truncation methods such as dissipation-assisted operator evolution (Rakovszky, von Keyserlingk, Pollmann, arXiv:2004.05177) converge empirically but carry no general error guarantee. Answer: an algorithm with controlled error, or evidence that controlled accuracy needs cost exponential in $1/\epsilon$.
What would settle it
An algorithm with certified error bars reproducing D to 1 percent in a benchmark chain, cross-checked by an independent method on the same model.
Status in the literature
Truncation methods since 2020 give mutually consistent D estimates in benchmark chains without a general error guarantee.