  Quantum circuits that are classically simulatable tell us when quantum
computation becomes less powerful than or equivalent to classical computation.
Such classically simulatable circuits are of importance because they illustrate
what makes universal quantum computation different from classical computers. In
this work, we propose a novel family of classically simulatable circuits by
making use of dual-unitary quantum circuits (DUQCs), which have been recently
investigated as exactly solvable models of non-equilibrium physics, and we
characterize their computational power. Specifically, we investigate the
computational complexity of the problem of calculating local expectation values
and the sampling problem of one-dimensional DUQCs, and we generalize them to
two spatial dimensions. We reveal that a local expectation value of a DUQC is
classically simulatable at an early time, which is linear in a system length.
In contrast, in a late time, they can perform universal quantum computation,
and the problem becomes a BQP-complete problem. Moreover, classical simulation
of sampling from a DUQC turns out to be hard.
