  Instantaneous quantum polynomial time (IQP) is a model of (probably)
non-universal quantum computation. Since it has been proven that IQP circuits
are unlikely to be simulated classically up to a multiplicative error and an
error in the $l_1$ norm, IQP is considered as one of the promising classes that
demonstrates quantum supremacy. Although IQP circuits can be realized more
easily than a universal quantum computer, demonstrating quantum supremacy is
still difficult. It is therefore desired to find subclasses of IQP that are
easy to implement. In this paper, by imposing some restrictions on IQP, we
propose ancilla-driven IQP (ADIQP) as the subclass of commuting quantum
computation suitable for many experimental settings. We show that even though
ADIQP circuits are strictly weaker than IQP circuits in a sense, they are also
hard to simulate classically up to a multiplicative error and an error in the
$l_1$ norm. Moreover, the properties of ADIQP make it easy to investigate the
verifiability of ADIQP circuits and the difficulties in realizing ADIQP
circuits.
