  Markov-chain Monte Carlo (MCMC), the field of stochastic algorithms built on
the concept of sampling, has countless applications in science and technology.
The overwhelming majority of MCMC algorithms are time-reversible and satisfy
the detailed-balance condition, just like physical systems in thermal
equilibrium. The underlying Markov chains typically display diffusive dynamics,
which leads to a slow exploration of sample space. Significant speed-ups can be
achieved by non-reversible MCMC algorithms exhibiting non-equilibrium dynamics,
whose steady states exactly reproduce the target equilibrium states of
reversible Markov chains. Such algorithms have had successes in applications
but are generally difficult to analyze, resulting in a scarcity of exact
results. Here, we introduce the "lifted" TASEP (totally asymmetric simple
exclusion process) as a paradigm for lifted non-reversible Markov chains. Our
model can be viewed as a second-generation lifting of the reversible Metropolis
algorithm on a one-dimensional lattice and is exactly solvable by an unusual
kind of coordinate Bethe ansatz. We establish the integrability of the model
and present strong evidence that the lifting leads to faster relaxation than in
the KPZ universality class.
