  Szegedy developed a generic method for quantizing classical algorithms based
on random walks [Proceedings of FOCS, 2004, pp. 32-41]. A major contribution of
his work was the construction of a walk unitary for any reversible random walk.
Such unitary posses two crucial properties: its eigenvector with eigenphase $0$
is a quantum sample of the limiting distribution of the random walk and its
eigenphase gap is quadratically larger than the spectral gap of the random
walk. It was an open question if it is possible to generalize Szegedy's
quantization method for stochastic maps to quantum maps. We answer this in the
affirmative by presenting an explicit construction of a Szegedy walk unitary
for detailed balanced Lindbladians -- generators of quantum Markov semigroups
-- and detailed balanced quantum channels. We prove that our Szegedy walk
unitary has a purification of the fixed point of the Lindbladian as eigenvector
with eigenphase $0$ and that its eigenphase gap is quadratically larger than
the spectral gap of the Lindbladian. To construct the walk unitary we leverage
a canonical form for detailed balanced Lindbladians showing that they are
structurally related to Davies generators. We also explain how the quantization
method for Lindbladians can be applied to quantum channels. We give an
efficient quantum algorithm for quantizing Davies generators that describe many
important open-system dynamics, for instance, the relaxation of a quantum
system coupled to a bath. Our algorithm extends known techniques for simulating
quantum systems on a quantum computer.
