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We show that finding the lowest eigenvalue of a 3-local symmetric stochasticmatrix is QMA-complete. We also show that finding the highest energy of astoquastic Hamiltonian is QMA-complete and that adiabatic quantum computationusing certain excited states of a stoquastic Hamiltonian is universal. We alsoshow that adiabatic evolution in the ground state of a stochastic frustrationfree Hamiltonian is universal. Our results give a new QMA-complete problemarising in the classical setting of Markov chains, and new adiabaticallyuniversal Hamiltonians that arise in many physical systems.

QMA-complete problems for stoquastic Hamiltonians and Markov matrices