  Sampling from the stationary distribution is one of the fundamental tasks of
Markov chain-based algorithms and has important applications in machine
learning, combinatorial optimization and network science. For the quantum case,
qsampling from Markov chains can be constructed as preparing quantum states
with amplitudes arbitrarily close to the square root of a stationary
distribution instead of classical sampling from a stationary distribution. In
this paper, a new qsampling algorithm for all reversible Markov chains is
constructed by discrete-time quantum walks and works without any limit compared
with existing results. In detail, we build a qsampling algorithm that not only
accelerates non-regular graphs but also keeps the speed-up of existing quantum
algorithms for regular graphs. In non-regular graphs, the invocation of the
quantum fast-forward algorithm accelerates existing state-of-the-art qsampling
algorithms for both discrete-time and continuous-time cases, especially on
sparse graphs. Compared to existing algorithms we reduce log n, where n is the
number of graph vertices. In regular graphs, our result matches other quantum
algorithms, and our reliance on the gap of Markov chains achieves quadratic
speedup compared with classical cases. For both cases, we reduce the number of
ancilla qubits required compared to the existing results. In some widely used
graphs and a series of sparse graphs where stationary distributions are
difficult to reach quickly, our algorithm is the first algorithm to achieve
complete quadratic acceleration (without log factor) over the classical case
without any limit. To enlarge success probability amplitude amplification is
introduced. We construct a new reflection on stationary state with fewer
ancilla qubits and think it may have independent application.
