  Diagonal quantum circuits are quantum circuits comprising only diagonal gates
in the computational basis. In spite of a classical feature of diagonal quantum
circuits in the sense of commutativity of all gates, their computational power
is highly likely to outperform classical one and they are exploited for
applications in quantum informational tasks. We review computational power of
diagonal quantum circuits and their applications. We focus on the computational
power of instantaneous quantum polynomial-time (IQP) circuits, which are a
special type of diagonal quantum circuits. We then review an approximate
generation of random states as an application of diagonal quantum circuits,
where random states are an ensemble of pure states uniformly distributed in a
Hilbert space. We also present a thermalizing algorithm of classical
Hamiltonians by using diagonal quantum circuits. These applications are
feasible to be experimentally implemented by current technology due to a simple
and robust structure of diagonal gates.
